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Theorem infmap2 9078
 Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 9436 avoids the axiom of choice, it requires the powerset of an infinite set to be well-orderable and so is usually not applicable. (Contributed by NM, 1-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
infmap2 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → (𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem infmap2
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 oveq2 6698 . . 3 (𝐵 = ∅ → (𝐴𝑚 𝐵) = (𝐴𝑚 ∅))
2 breq2 4689 . . . . 5 (𝐵 = ∅ → (𝑥𝐵𝑥 ≈ ∅))
32anbi2d 740 . . . 4 (𝐵 = ∅ → ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑥 ≈ ∅)))
43abbidv 2770 . . 3 (𝐵 = ∅ → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)})
51, 4breq12d 4698 . 2 (𝐵 = ∅ → ((𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ↔ (𝐴𝑚 ∅) ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)}))
6 simpl2 1085 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵𝐴)
7 reldom 8003 . . . . . . . . . . 11 Rel ≼
87brrelexi 5192 . . . . . . . . . 10 (𝐵𝐴𝐵 ∈ V)
96, 8syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ∈ V)
107brrelex2i 5193 . . . . . . . . . 10 (𝐵𝐴𝐴 ∈ V)
116, 10syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ V)
12 xpcomeng 8093 . . . . . . . . 9 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
139, 11, 12syl2anc 694 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
14 simpl3 1086 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴𝑚 𝐵) ∈ dom card)
15 simpr 476 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅)
16 mapdom3 8173 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴𝑚 𝐵))
1711, 9, 15, 16syl3anc 1366 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴𝑚 𝐵))
18 numdom 8899 . . . . . . . . . 10 (((𝐴𝑚 𝐵) ∈ dom card ∧ 𝐴 ≼ (𝐴𝑚 𝐵)) → 𝐴 ∈ dom card)
1914, 17, 18syl2anc 694 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ dom card)
20 simpl1 1084 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ω ≼ 𝐴)
21 infxpabs 9072 . . . . . . . . 9 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
2219, 20, 15, 6, 21syl22anc 1367 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 × 𝐵) ≈ 𝐴)
23 entr 8049 . . . . . . . 8 (((𝐵 × 𝐴) ≈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ≈ 𝐴) → (𝐵 × 𝐴) ≈ 𝐴)
2413, 22, 23syl2anc 694 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ 𝐴)
25 ssenen 8175 . . . . . . 7 ((𝐵 × 𝐴) ≈ 𝐴 → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
2624, 25syl 17 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
27 relen 8002 . . . . . . 7 Rel ≈
2827brrelexi 5192 . . . . . 6 ({𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ∈ V)
2926, 28syl 17 . . . . 5 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ∈ V)
30 abid2 2774 . . . . . 6 {𝑥𝑥 ∈ (𝐴𝑚 𝐵)} = (𝐴𝑚 𝐵)
31 elmapi 7921 . . . . . . . 8 (𝑥 ∈ (𝐴𝑚 𝐵) → 𝑥:𝐵𝐴)
32 fssxp 6098 . . . . . . . . 9 (𝑥:𝐵𝐴𝑥 ⊆ (𝐵 × 𝐴))
33 ffun 6086 . . . . . . . . . . 11 (𝑥:𝐵𝐴 → Fun 𝑥)
34 vex 3234 . . . . . . . . . . . 12 𝑥 ∈ V
3534fundmen 8071 . . . . . . . . . . 11 (Fun 𝑥 → dom 𝑥𝑥)
36 ensym 8046 . . . . . . . . . . 11 (dom 𝑥𝑥𝑥 ≈ dom 𝑥)
3733, 35, 363syl 18 . . . . . . . . . 10 (𝑥:𝐵𝐴𝑥 ≈ dom 𝑥)
38 fdm 6089 . . . . . . . . . 10 (𝑥:𝐵𝐴 → dom 𝑥 = 𝐵)
3937, 38breqtrd 4711 . . . . . . . . 9 (𝑥:𝐵𝐴𝑥𝐵)
4032, 39jca 553 . . . . . . . 8 (𝑥:𝐵𝐴 → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵))
4131, 40syl 17 . . . . . . 7 (𝑥 ∈ (𝐴𝑚 𝐵) → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵))
4241ss2abi 3707 . . . . . 6 {𝑥𝑥 ∈ (𝐴𝑚 𝐵)} ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)}
4330, 42eqsstr3i 3669 . . . . 5 (𝐴𝑚 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)}
44 ssdomg 8043 . . . . 5 ({𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ∈ V → ((𝐴𝑚 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} → (𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)}))
4529, 43, 44mpisyl 21 . . . 4 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)})
46 domentr 8056 . . . 4 (((𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ∧ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)}) → (𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
4745, 26, 46syl2anc 694 . . 3 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
48 ovex 6718 . . . . . . 7 (𝐴𝑚 𝐵) ∈ V
4948mptex 6527 . . . . . 6 (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ∈ V
5049rnex 7142 . . . . 5 ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ∈ V
51 ensym 8046 . . . . . . . . . . . 12 (𝑥𝐵𝐵𝑥)
5251ad2antll 765 . . . . . . . . . . 11 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → 𝐵𝑥)
53 bren 8006 . . . . . . . . . . 11 (𝐵𝑥 ↔ ∃𝑓 𝑓:𝐵1-1-onto𝑥)
5452, 53sylib 208 . . . . . . . . . 10 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → ∃𝑓 𝑓:𝐵1-1-onto𝑥)
55 f1of 6175 . . . . . . . . . . . . . . . 16 (𝑓:𝐵1-1-onto𝑥𝑓:𝐵𝑥)
5655adantl 481 . . . . . . . . . . . . . . 15 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓:𝐵𝑥)
57 simplrl 817 . . . . . . . . . . . . . . 15 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑥𝐴)
5856, 57fssd 6095 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓:𝐵𝐴)
5911, 9elmapd 7913 . . . . . . . . . . . . . . 15 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴𝑚 𝐵) ↔ 𝑓:𝐵𝐴))
6059ad2antrr 762 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → (𝑓 ∈ (𝐴𝑚 𝐵) ↔ 𝑓:𝐵𝐴))
6158, 60mpbird 247 . . . . . . . . . . . . 13 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓 ∈ (𝐴𝑚 𝐵))
62 f1ofo 6182 . . . . . . . . . . . . . . . 16 (𝑓:𝐵1-1-onto𝑥𝑓:𝐵onto𝑥)
63 forn 6156 . . . . . . . . . . . . . . . 16 (𝑓:𝐵onto𝑥 → ran 𝑓 = 𝑥)
6462, 63syl 17 . . . . . . . . . . . . . . 15 (𝑓:𝐵1-1-onto𝑥 → ran 𝑓 = 𝑥)
6564adantl 481 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → ran 𝑓 = 𝑥)
6665eqcomd 2657 . . . . . . . . . . . . 13 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑥 = ran 𝑓)
6761, 66jca 553 . . . . . . . . . . . 12 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → (𝑓 ∈ (𝐴𝑚 𝐵) ∧ 𝑥 = ran 𝑓))
6867ex 449 . . . . . . . . . . 11 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → (𝑓:𝐵1-1-onto𝑥 → (𝑓 ∈ (𝐴𝑚 𝐵) ∧ 𝑥 = ran 𝑓)))
6968eximdv 1886 . . . . . . . . . 10 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → (∃𝑓 𝑓:𝐵1-1-onto𝑥 → ∃𝑓(𝑓 ∈ (𝐴𝑚 𝐵) ∧ 𝑥 = ran 𝑓)))
7054, 69mpd 15 . . . . . . . . 9 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → ∃𝑓(𝑓 ∈ (𝐴𝑚 𝐵) ∧ 𝑥 = ran 𝑓))
71 df-rex 2947 . . . . . . . . 9 (∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓 ↔ ∃𝑓(𝑓 ∈ (𝐴𝑚 𝐵) ∧ 𝑥 = ran 𝑓))
7270, 71sylibr 224 . . . . . . . 8 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → ∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓)
7372ex 449 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ((𝑥𝐴𝑥𝐵) → ∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓))
7473ss2abdv 3708 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ⊆ {𝑥 ∣ ∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓})
75 eqid 2651 . . . . . . 7 (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) = (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓)
7675rnmpt 5403 . . . . . 6 ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) = {𝑥 ∣ ∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓}
7774, 76syl6sseqr 3685 . . . . 5 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ⊆ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓))
78 ssdomg 8043 . . . . 5 (ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ∈ V → ({𝑥 ∣ (𝑥𝐴𝑥𝐵)} ⊆ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓)))
7950, 77, 78mpsyl 68 . . . 4 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓))
80 vex 3234 . . . . . . . . 9 𝑓 ∈ V
8180rnex 7142 . . . . . . . 8 ran 𝑓 ∈ V
8281rgenw 2953 . . . . . . 7 𝑓 ∈ (𝐴𝑚 𝐵)ran 𝑓 ∈ V
8375fnmpt 6058 . . . . . . 7 (∀𝑓 ∈ (𝐴𝑚 𝐵)ran 𝑓 ∈ V → (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) Fn (𝐴𝑚 𝐵))
8482, 83mp1i 13 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) Fn (𝐴𝑚 𝐵))
85 dffn4 6159 . . . . . 6 ((𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) Fn (𝐴𝑚 𝐵) ↔ (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓):(𝐴𝑚 𝐵)–onto→ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓))
8684, 85sylib 208 . . . . 5 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓):(𝐴𝑚 𝐵)–onto→ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓))
87 fodomnum 8918 . . . . 5 ((𝐴𝑚 𝐵) ∈ dom card → ((𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓):(𝐴𝑚 𝐵)–onto→ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) → ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ≼ (𝐴𝑚 𝐵)))
8814, 86, 87sylc 65 . . . 4 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ≼ (𝐴𝑚 𝐵))
89 domtr 8050 . . . 4 (({𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ∧ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ≼ (𝐴𝑚 𝐵)) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴𝑚 𝐵))
9079, 88, 89syl2anc 694 . . 3 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴𝑚 𝐵))
91 sbth 8121 . . 3 (((𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ∧ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴𝑚 𝐵)) → (𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
9247, 90, 91syl2anc 694 . 2 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
937brrelex2i 5193 . . . . 5 (ω ≼ 𝐴𝐴 ∈ V)
94933ad2ant1 1102 . . . 4 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → 𝐴 ∈ V)
95 map0e 7937 . . . 4 (𝐴 ∈ V → (𝐴𝑚 ∅) = 1𝑜)
9694, 95syl 17 . . 3 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → (𝐴𝑚 ∅) = 1𝑜)
97 1onn 7764 . . . . . 6 1𝑜 ∈ ω
9897elexi 3244 . . . . 5 1𝑜 ∈ V
9998enref 8030 . . . 4 1𝑜 ≈ 1𝑜
100 df-sn 4211 . . . . 5 {∅} = {𝑥𝑥 = ∅}
101 df1o2 7617 . . . . 5 1𝑜 = {∅}
102 en0 8060 . . . . . . . 8 (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
103102anbi2i 730 . . . . . . 7 ((𝑥𝐴𝑥 ≈ ∅) ↔ (𝑥𝐴𝑥 = ∅))
104 0ss 4005 . . . . . . . . 9 ∅ ⊆ 𝐴
105 sseq1 3659 . . . . . . . . 9 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ⊆ 𝐴))
106104, 105mpbiri 248 . . . . . . . 8 (𝑥 = ∅ → 𝑥𝐴)
107106pm4.71ri 666 . . . . . . 7 (𝑥 = ∅ ↔ (𝑥𝐴𝑥 = ∅))
108103, 107bitr4i 267 . . . . . 6 ((𝑥𝐴𝑥 ≈ ∅) ↔ 𝑥 = ∅)
109108abbii 2768 . . . . 5 {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)} = {𝑥𝑥 = ∅}
110100, 101, 1093eqtr4ri 2684 . . . 4 {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)} = 1𝑜
11199, 110breqtrri 4712 . . 3 1𝑜 ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)}
11296, 111syl6eqbr 4724 . 2 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → (𝐴𝑚 ∅) ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)})
1135, 92, 112pm2.61ne 2908 1 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → (𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030  {cab 2637   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ⊆ wss 3607  ∅c0 3948  {csn 4210   class class class wbr 4685   ↦ cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  Fun wfun 5920   Fn wfn 5921  ⟶wf 5922  –onto→wfo 5924  –1-1-onto→wf1o 5925  (class class class)co 6690  ωcom 7107  1𝑜c1o 7598   ↑𝑚 cmap 7899   ≈ cen 7994   ≼ cdom 7995  cardccrd 8799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-card 8803  df-acn 8806 This theorem is referenced by:  infmap  9436
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