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Theorem infmap2 8900
Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 9254 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 6534 . . 3 (𝐵 = ∅ → (𝐴𝑚 𝐵) = (𝐴𝑚 ∅))
2 breq2 4581 . . . . 5 (𝐵 = ∅ → (𝑥𝐵𝑥 ≈ ∅))
32anbi2d 735 . . . 4 (𝐵 = ∅ → ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑥 ≈ ∅)))
43abbidv 2727 . . 3 (𝐵 = ∅ → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)})
51, 4breq12d 4590 . 2 (𝐵 = ∅ → ((𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ↔ (𝐴𝑚 ∅) ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)}))
6 simpl2 1057 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵𝐴)
7 reldom 7824 . . . . . . . . . . 11 Rel ≼
87brrelexi 5071 . . . . . . . . . 10 (𝐵𝐴𝐵 ∈ V)
96, 8syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ∈ V)
107brrelex2i 5072 . . . . . . . . . 10 (𝐵𝐴𝐴 ∈ V)
116, 10syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ V)
12 xpcomeng 7914 . . . . . . . . 9 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
139, 11, 12syl2anc 690 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
14 simpl3 1058 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴𝑚 𝐵) ∈ dom card)
15 simpr 475 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅)
16 mapdom3 7994 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴𝑚 𝐵))
1711, 9, 15, 16syl3anc 1317 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴𝑚 𝐵))
18 numdom 8721 . . . . . . . . . 10 (((𝐴𝑚 𝐵) ∈ dom card ∧ 𝐴 ≼ (𝐴𝑚 𝐵)) → 𝐴 ∈ dom card)
1914, 17, 18syl2anc 690 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ dom card)
20 simpl1 1056 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ω ≼ 𝐴)
21 infxpabs 8894 . . . . . . . . 9 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
2219, 20, 15, 6, 21syl22anc 1318 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 × 𝐵) ≈ 𝐴)
23 entr 7871 . . . . . . . 8 (((𝐵 × 𝐴) ≈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ≈ 𝐴) → (𝐵 × 𝐴) ≈ 𝐴)
2413, 22, 23syl2anc 690 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ 𝐴)
25 ssenen 7996 . . . . . . 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 7823 . . . . . . 7 Rel ≈
2827brrelexi 5071 . . . . . 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 2731 . . . . . 6 {𝑥𝑥 ∈ (𝐴𝑚 𝐵)} = (𝐴𝑚 𝐵)
31 elmapi 7742 . . . . . . . 8 (𝑥 ∈ (𝐴𝑚 𝐵) → 𝑥:𝐵𝐴)
32 fssxp 5958 . . . . . . . . 9 (𝑥:𝐵𝐴𝑥 ⊆ (𝐵 × 𝐴))
33 ffun 5946 . . . . . . . . . . 11 (𝑥:𝐵𝐴 → Fun 𝑥)
34 vex 3175 . . . . . . . . . . . 12 𝑥 ∈ V
3534fundmen 7893 . . . . . . . . . . 11 (Fun 𝑥 → dom 𝑥𝑥)
36 ensym 7868 . . . . . . . . . . 11 (dom 𝑥𝑥𝑥 ≈ dom 𝑥)
3733, 35, 363syl 18 . . . . . . . . . 10 (𝑥:𝐵𝐴𝑥 ≈ dom 𝑥)
38 fdm 5949 . . . . . . . . . 10 (𝑥:𝐵𝐴 → dom 𝑥 = 𝐵)
3937, 38breqtrd 4603 . . . . . . . . 9 (𝑥:𝐵𝐴𝑥𝐵)
4032, 39jca 552 . . . . . . . 8 (𝑥:𝐵𝐴 → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵))
4131, 40syl 17 . . . . . . 7 (𝑥 ∈ (𝐴𝑚 𝐵) → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵))
4241ss2abi 3636 . . . . . 6 {𝑥𝑥 ∈ (𝐴𝑚 𝐵)} ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)}
4330, 42eqsstr3i 3598 . . . . 5 (𝐴𝑚 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)}
44 ssdomg 7864 . . . . 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 7878 . . . 4 (((𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ∧ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)}) → (𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
4745, 26, 46syl2anc 690 . . 3 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
48 ovex 6554 . . . . . . 7 (𝐴𝑚 𝐵) ∈ V
4948mptex 6367 . . . . . 6 (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ∈ V
5049rnex 6969 . . . . 5 ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ∈ V
51 ensym 7868 . . . . . . . . . . . 12 (𝑥𝐵𝐵𝑥)
5251ad2antll 760 . . . . . . . . . . 11 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → 𝐵𝑥)
53 bren 7827 . . . . . . . . . . 11 (𝐵𝑥 ↔ ∃𝑓 𝑓:𝐵1-1-onto𝑥)
5452, 53sylib 206 . . . . . . . . . 10 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → ∃𝑓 𝑓:𝐵1-1-onto𝑥)
55 f1of 6034 . . . . . . . . . . . . . . . 16 (𝑓:𝐵1-1-onto𝑥𝑓:𝐵𝑥)
5655adantl 480 . . . . . . . . . . . . . . 15 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓:𝐵𝑥)
57 simplrl 795 . . . . . . . . . . . . . . 15 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑥𝐴)
5856, 57fssd 5955 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓:𝐵𝐴)
5911, 9elmapd 7735 . . . . . . . . . . . . . . 15 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴𝑚 𝐵) ↔ 𝑓:𝐵𝐴))
6059ad2antrr 757 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → (𝑓 ∈ (𝐴𝑚 𝐵) ↔ 𝑓:𝐵𝐴))
6158, 60mpbird 245 . . . . . . . . . . . . 13 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓 ∈ (𝐴𝑚 𝐵))
62 f1ofo 6041 . . . . . . . . . . . . . . . 16 (𝑓:𝐵1-1-onto𝑥𝑓:𝐵onto𝑥)
63 forn 6015 . . . . . . . . . . . . . . . 16 (𝑓:𝐵onto𝑥 → ran 𝑓 = 𝑥)
6462, 63syl 17 . . . . . . . . . . . . . . 15 (𝑓:𝐵1-1-onto𝑥 → ran 𝑓 = 𝑥)
6564adantl 480 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → ran 𝑓 = 𝑥)
6665eqcomd 2615 . . . . . . . . . . . . 13 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑥 = ran 𝑓)
6761, 66jca 552 . . . . . . . . . . . 12 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → (𝑓 ∈ (𝐴𝑚 𝐵) ∧ 𝑥 = ran 𝑓))
6867ex 448 . . . . . . . . . . 11 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → (𝑓:𝐵1-1-onto𝑥 → (𝑓 ∈ (𝐴𝑚 𝐵) ∧ 𝑥 = ran 𝑓)))
6968eximdv 1832 . . . . . . . . . 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 2901 . . . . . . . . 9 (∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓 ↔ ∃𝑓(𝑓 ∈ (𝐴𝑚 𝐵) ∧ 𝑥 = ran 𝑓))
7270, 71sylibr 222 . . . . . . . 8 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → ∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓)
7372ex 448 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ((𝑥𝐴𝑥𝐵) → ∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓))
7473ss2abdv 3637 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ⊆ {𝑥 ∣ ∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓})
75 eqid 2609 . . . . . . 7 (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) = (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓)
7675rnmpt 5278 . . . . . 6 ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) = {𝑥 ∣ ∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓}
7774, 76syl6sseqr 3614 . . . . 5 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ⊆ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓))
78 ssdomg 7864 . . . . 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 65 . . . 4 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓))
80 vex 3175 . . . . . . . . 9 𝑓 ∈ V
8180rnex 6969 . . . . . . . 8 ran 𝑓 ∈ V
8281rgenw 2907 . . . . . . 7 𝑓 ∈ (𝐴𝑚 𝐵)ran 𝑓 ∈ V
8375fnmpt 5918 . . . . . . 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 6018 . . . . . 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 206 . . . . 5 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓):(𝐴𝑚 𝐵)–onto→ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓))
87 fodomnum 8740 . . . . 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 62 . . . 4 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ≼ (𝐴𝑚 𝐵))
89 domtr 7872 . . . 4 (({𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ∧ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ≼ (𝐴𝑚 𝐵)) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴𝑚 𝐵))
9079, 88, 89syl2anc 690 . . 3 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴𝑚 𝐵))
91 sbth 7942 . . 3 (((𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ∧ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴𝑚 𝐵)) → (𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
9247, 90, 91syl2anc 690 . 2 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
937brrelex2i 5072 . . . . 5 (ω ≼ 𝐴𝐴 ∈ V)
94933ad2ant1 1074 . . . 4 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → 𝐴 ∈ V)
95 map0e 7758 . . . 4 (𝐴 ∈ V → (𝐴𝑚 ∅) = 1𝑜)
9694, 95syl 17 . . 3 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → (𝐴𝑚 ∅) = 1𝑜)
97 1onn 7583 . . . . . 6 1𝑜 ∈ ω
9897elexi 3185 . . . . 5 1𝑜 ∈ V
9998enref 7851 . . . 4 1𝑜 ≈ 1𝑜
100 df-sn 4125 . . . . 5 {∅} = {𝑥𝑥 = ∅}
101 df1o2 7436 . . . . 5 1𝑜 = {∅}
102 en0 7882 . . . . . . . 8 (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
103102anbi2i 725 . . . . . . 7 ((𝑥𝐴𝑥 ≈ ∅) ↔ (𝑥𝐴𝑥 = ∅))
104 0ss 3923 . . . . . . . . 9 ∅ ⊆ 𝐴
105 sseq1 3588 . . . . . . . . 9 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ⊆ 𝐴))
106104, 105mpbiri 246 . . . . . . . 8 (𝑥 = ∅ → 𝑥𝐴)
107106pm4.71ri 662 . . . . . . 7 (𝑥 = ∅ ↔ (𝑥𝐴𝑥 = ∅))
108103, 107bitr4i 265 . . . . . 6 ((𝑥𝐴𝑥 ≈ ∅) ↔ 𝑥 = ∅)
109108abbii 2725 . . . . 5 {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)} = {𝑥𝑥 = ∅}
110100, 101, 1093eqtr4ri 2642 . . . 4 {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)} = 1𝑜
11199, 110breqtrri 4604 . . 3 1𝑜 ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)}
11296, 111syl6eqbr 4616 . 2 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → (𝐴𝑚 ∅) ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)})
1135, 92, 112pm2.61ne 2866 1 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → (𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1976  {cab 2595  wne 2779  wral 2895  wrex 2896  Vcvv 3172  wss 3539  c0 3873  {csn 4124   class class class wbr 4577  cmpt 4637   × cxp 5025  dom cdm 5027  ran crn 5028  Fun wfun 5783   Fn wfn 5784  wf 5785  ontowfo 5787  1-1-ontowf1o 5788  (class class class)co 6526  ωcom 6934  1𝑜c1o 7417  𝑚 cmap 7721  cen 7815  cdom 7816  cardccrd 8621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-inf2 8398
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-se 4987  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-isom 5798  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-oi 8275  df-card 8625  df-acn 8628
This theorem is referenced by:  infmap  9254
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