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Theorem domssex 8065
Description: Weakening of domssex 8065 to forget the functions in favor of dominance and equinumerosity. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
domssex (𝐴𝐵 → ∃𝑥(𝐴𝑥𝐵𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem domssex
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 brdomi 7910 . 2 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
2 reldom 7905 . . 3 Rel ≼
32brrelex2i 5119 . 2 (𝐴𝐵𝐵 ∈ V)
4 vex 3189 . . . . . . . 8 𝑓 ∈ V
5 f1stres 7135 . . . . . . . . . 10 (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})⟶(𝐵 ∖ ran 𝑓)
65a1i 11 . . . . . . . . 9 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})⟶(𝐵 ∖ ran 𝑓))
7 difexg 4768 . . . . . . . . . . 11 (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
87adantl 482 . . . . . . . . . 10 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝐵 ∖ ran 𝑓) ∈ V)
9 snex 4869 . . . . . . . . . 10 {𝒫 ran 𝐴} ∈ V
10 xpexg 6913 . . . . . . . . . 10 (((𝐵 ∖ ran 𝑓) ∈ V ∧ {𝒫 ran 𝐴} ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}) ∈ V)
118, 9, 10sylancl 693 . . . . . . . . 9 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}) ∈ V)
12 fex2 7068 . . . . . . . . 9 (((1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})⟶(𝐵 ∖ ran 𝑓) ∧ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}) ∈ V ∧ (𝐵 ∖ ran 𝑓) ∈ V) → (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})) ∈ V)
136, 11, 8, 12syl3anc 1323 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})) ∈ V)
14 unexg 6912 . . . . . . . 8 ((𝑓 ∈ V ∧ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})) ∈ V) → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
154, 13, 14sylancr 694 . . . . . . 7 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
16 cnvexg 7059 . . . . . . 7 ((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
1715, 16syl 17 . . . . . 6 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
18 rnexg 7045 . . . . . 6 ((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V → ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
1917, 18syl 17 . . . . 5 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
20 simpl 473 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝑓:𝐴1-1𝐵)
21 f1dm 6062 . . . . . . . . . 10 (𝑓:𝐴1-1𝐵 → dom 𝑓 = 𝐴)
224dmex 7046 . . . . . . . . . 10 dom 𝑓 ∈ V
2321, 22syl6eqelr 2707 . . . . . . . . 9 (𝑓:𝐴1-1𝐵𝐴 ∈ V)
2423adantr 481 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐴 ∈ V)
25 simpr 477 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐵 ∈ V)
26 eqid 2621 . . . . . . . . 9 (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) = (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))
2726domss2 8063 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))):𝐵1-1-onto→ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∧ 𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∧ ((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∘ 𝑓) = ( I ↾ 𝐴)))
2820, 24, 25, 27syl3anc 1323 . . . . . . 7 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → ((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))):𝐵1-1-onto→ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∧ 𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∧ ((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∘ 𝑓) = ( I ↾ 𝐴)))
2928simp2d 1072 . . . . . 6 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))))
3028simp1d 1071 . . . . . . 7 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))):𝐵1-1-onto→ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))))
31 f1oen3g 7915 . . . . . . 7 (((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V ∧ (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))):𝐵1-1-onto→ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))) → 𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))))
3217, 30, 31syl2anc 692 . . . . . 6 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))))
3329, 32jca 554 . . . . 5 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∧ 𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))))
34 sseq2 3606 . . . . . . 7 (𝑥 = ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) → (𝐴𝑥𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))))
35 breq2 4617 . . . . . . 7 (𝑥 = ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) → (𝐵𝑥𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))))
3634, 35anbi12d 746 . . . . . 6 (𝑥 = ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) → ((𝐴𝑥𝐵𝑥) ↔ (𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∧ 𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))))))
3736spcegv 3280 . . . . 5 (ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V → ((𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∧ 𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))) → ∃𝑥(𝐴𝑥𝐵𝑥)))
3819, 33, 37sylc 65 . . . 4 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → ∃𝑥(𝐴𝑥𝐵𝑥))
3938ex 450 . . 3 (𝑓:𝐴1-1𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴𝑥𝐵𝑥)))
4039exlimiv 1855 . 2 (∃𝑓 𝑓:𝐴1-1𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴𝑥𝐵𝑥)))
411, 3, 40sylc 65 1 (𝐴𝐵 → ∃𝑥(𝐴𝑥𝐵𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  cdif 3552  cun 3553  wss 3555  𝒫 cpw 4130  {csn 4148   cuni 4402   class class class wbr 4613   I cid 4984   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  cres 5076  ccom 5078  wf 5843  1-1wf1 5844  1-1-ontowf1o 5846  1st c1st 7111  cen 7896  cdom 7897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-1st 7113  df-2nd 7114  df-en 7900  df-dom 7901
This theorem is referenced by: (None)
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