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Theorem infmap2 10257
Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 10616 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 7439 . . 3 (𝐵 = ∅ → (𝐴m 𝐵) = (𝐴m ∅))
2 breq2 5147 . . . . 5 (𝐵 = ∅ → (𝑥𝐵𝑥 ≈ ∅))
32anbi2d 630 . . . 4 (𝐵 = ∅ → ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑥 ≈ ∅)))
43abbidv 2808 . . 3 (𝐵 = ∅ → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)})
51, 4breq12d 5156 . 2 (𝐵 = ∅ → ((𝐴m 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ↔ (𝐴m ∅) ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)}))
6 simpl2 1193 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵𝐴)
7 reldom 8991 . . . . . . . . . . 11 Rel ≼
87brrelex1i 5741 . . . . . . . . . 10 (𝐵𝐴𝐵 ∈ V)
96, 8syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ∈ V)
107brrelex2i 5742 . . . . . . . . . 10 (𝐵𝐴𝐴 ∈ V)
116, 10syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ V)
12 xpcomeng 9104 . . . . . . . . 9 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
139, 11, 12syl2anc 584 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
14 simpl3 1194 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴m 𝐵) ∈ dom card)
15 simpr 484 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅)
16 mapdom3 9189 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴m 𝐵))
1711, 9, 15, 16syl3anc 1373 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴m 𝐵))
18 numdom 10078 . . . . . . . . . 10 (((𝐴m 𝐵) ∈ dom card ∧ 𝐴 ≼ (𝐴m 𝐵)) → 𝐴 ∈ dom card)
1914, 17, 18syl2anc 584 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ dom card)
20 simpl1 1192 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ω ≼ 𝐴)
21 infxpabs 10251 . . . . . . . . 9 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
2219, 20, 15, 6, 21syl22anc 839 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 × 𝐵) ≈ 𝐴)
23 entr 9046 . . . . . . . 8 (((𝐵 × 𝐴) ≈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ≈ 𝐴) → (𝐵 × 𝐴) ≈ 𝐴)
2413, 22, 23syl2anc 584 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ 𝐴)
25 ssenen 9191 . . . . . . 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 8990 . . . . . . 7 Rel ≈
2827brrelex1i 5741 . . . . . 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 2879 . . . . . 6 {𝑥𝑥 ∈ (𝐴m 𝐵)} = (𝐴m 𝐵)
31 elmapi 8889 . . . . . . . 8 (𝑥 ∈ (𝐴m 𝐵) → 𝑥:𝐵𝐴)
32 fssxp 6763 . . . . . . . . 9 (𝑥:𝐵𝐴𝑥 ⊆ (𝐵 × 𝐴))
33 ffun 6739 . . . . . . . . . . 11 (𝑥:𝐵𝐴 → Fun 𝑥)
34 vex 3484 . . . . . . . . . . . 12 𝑥 ∈ V
3534fundmen 9071 . . . . . . . . . . 11 (Fun 𝑥 → dom 𝑥𝑥)
36 ensym 9043 . . . . . . . . . . 11 (dom 𝑥𝑥𝑥 ≈ dom 𝑥)
3733, 35, 363syl 18 . . . . . . . . . 10 (𝑥:𝐵𝐴𝑥 ≈ dom 𝑥)
38 fdm 6745 . . . . . . . . . 10 (𝑥:𝐵𝐴 → dom 𝑥 = 𝐵)
3937, 38breqtrd 5169 . . . . . . . . 9 (𝑥:𝐵𝐴𝑥𝐵)
4032, 39jca 511 . . . . . . . 8 (𝑥:𝐵𝐴 → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵))
4131, 40syl 17 . . . . . . 7 (𝑥 ∈ (𝐴m 𝐵) → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵))
4241ss2abi 4067 . . . . . 6 {𝑥𝑥 ∈ (𝐴m 𝐵)} ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)}
4330, 42eqsstrri 4031 . . . . 5 (𝐴m 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)}
44 ssdomg 9040 . . . . 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 9053 . . . 4 (((𝐴m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ∧ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)}) → (𝐴m 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
4745, 26, 46syl2anc 584 . . 3 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴m 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
48 ovex 7464 . . . . . . 7 (𝐴m 𝐵) ∈ V
4948mptex 7243 . . . . . 6 (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) ∈ V
5049rnex 7932 . . . . 5 ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) ∈ V
51 ensym 9043 . . . . . . . . . . . 12 (𝑥𝐵𝐵𝑥)
5251ad2antll 729 . . . . . . . . . . 11 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → 𝐵𝑥)
53 bren 8995 . . . . . . . . . . 11 (𝐵𝑥 ↔ ∃𝑓 𝑓:𝐵1-1-onto𝑥)
5452, 53sylib 218 . . . . . . . . . 10 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → ∃𝑓 𝑓:𝐵1-1-onto𝑥)
55 f1of 6848 . . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . 15 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑥𝐴)
5856, 57fssd 6753 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓:𝐵𝐴)
5911, 9elmapd 8880 . . . . . . . . . . . . . . 15 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴m 𝐵) ↔ 𝑓:𝐵𝐴))
6059ad2antrr 726 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → (𝑓 ∈ (𝐴m 𝐵) ↔ 𝑓:𝐵𝐴))
6158, 60mpbird 257 . . . . . . . . . . . . 13 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓 ∈ (𝐴m 𝐵))
62 f1ofo 6855 . . . . . . . . . . . . . . . 16 (𝑓:𝐵1-1-onto𝑥𝑓:𝐵onto𝑥)
63 forn 6823 . . . . . . . . . . . . . . . 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 2743 . . . . . . . . . . . . 13 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑥 = ran 𝑓)
6761, 66jca 511 . . . . . . . . . . . 12 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → (𝑓 ∈ (𝐴m 𝐵) ∧ 𝑥 = ran 𝑓))
6867ex 412 . . . . . . . . . . 11 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → (𝑓:𝐵1-1-onto𝑥 → (𝑓 ∈ (𝐴m 𝐵) ∧ 𝑥 = ran 𝑓)))
6968eximdv 1917 . . . . . . . . . 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 3071 . . . . . . . . 9 (∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓 ↔ ∃𝑓(𝑓 ∈ (𝐴m 𝐵) ∧ 𝑥 = ran 𝑓))
7270, 71sylibr 234 . . . . . . . 8 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → ∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓)
7372ex 412 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ((𝑥𝐴𝑥𝐵) → ∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓))
7473ss2abdv 4066 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ⊆ {𝑥 ∣ ∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓})
75 eqid 2737 . . . . . . 7 (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) = (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓)
7675rnmpt 5968 . . . . . 6 ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) = {𝑥 ∣ ∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓}
7774, 76sseqtrrdi 4025 . . . . 5 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ⊆ ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓))
78 ssdomg 9040 . . . . 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 3484 . . . . . . . . 9 𝑓 ∈ V
8180rnex 7932 . . . . . . . 8 ran 𝑓 ∈ V
8281rgenw 3065 . . . . . . 7 𝑓 ∈ (𝐴m 𝐵)ran 𝑓 ∈ V
8375fnmpt 6708 . . . . . . 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 6826 . . . . . 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 218 . . . . 5 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓):(𝐴m 𝐵)–onto→ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓))
87 fodomnum 10097 . . . . 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 9047 . . . 4 (({𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) ∧ ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) ≼ (𝐴m 𝐵)) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴m 𝐵))
9079, 88, 89syl2anc 584 . . 3 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴m 𝐵))
91 sbth 9133 . . 3 (((𝐴m 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ∧ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴m 𝐵)) → (𝐴m 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
9247, 90, 91syl2anc 584 . 2 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴m 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
937brrelex2i 5742 . . . . 5 (ω ≼ 𝐴𝐴 ∈ V)
94933ad2ant1 1134 . . . 4 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) → 𝐴 ∈ V)
95 map0e 8922 . . . 4 (𝐴 ∈ V → (𝐴m ∅) = 1o)
9694, 95syl 17 . . 3 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) → (𝐴m ∅) = 1o)
97 1oex 8516 . . . . 5 1o ∈ V
9897enref 9025 . . . 4 1o ≈ 1o
99 df-sn 4627 . . . . 5 {∅} = {𝑥𝑥 = ∅}
100 df1o2 8513 . . . . 5 1o = {∅}
101 en0 9058 . . . . . . . 8 (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
102101anbi2i 623 . . . . . . 7 ((𝑥𝐴𝑥 ≈ ∅) ↔ (𝑥𝐴𝑥 = ∅))
103 0ss 4400 . . . . . . . . 9 ∅ ⊆ 𝐴
104 sseq1 4009 . . . . . . . . 9 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ⊆ 𝐴))
105103, 104mpbiri 258 . . . . . . . 8 (𝑥 = ∅ → 𝑥𝐴)
106105pm4.71ri 560 . . . . . . 7 (𝑥 = ∅ ↔ (𝑥𝐴𝑥 = ∅))
107102, 106bitr4i 278 . . . . . 6 ((𝑥𝐴𝑥 ≈ ∅) ↔ 𝑥 = ∅)
108107abbii 2809 . . . . 5 {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)} = {𝑥𝑥 = ∅}
10999, 100, 1083eqtr4ri 2776 . . . 4 {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)} = 1o
11098, 109breqtrri 5170 . . 3 1o ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)}
11196, 110eqbrtrdi 5182 . 2 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) → (𝐴m ∅) ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)})
1125, 92, 111pm2.61ne 3027 1 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) → (𝐴m 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wne 2940  wral 3061  wrex 3070  Vcvv 3480  wss 3951  c0 4333  {csn 4626   class class class wbr 5143  cmpt 5225   × cxp 5683  dom cdm 5685  ran crn 5686  Fun wfun 6555   Fn wfn 6556  wf 6557  ontowfo 6559  1-1-ontowf1o 6560  (class class class)co 7431  ωcom 7887  1oc1o 8499  m cmap 8866  cen 8982  cdom 8983  cardccrd 9975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-card 9979  df-acn 9982
This theorem is referenced by:  infmap  10616
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