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Theorem domssex 8474
 Description: Weakening of domssex 8474 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 8317 . 2 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
2 reldom 8312 . . 3 Rel ≼
32brrelex2i 5459 . 2 (𝐴𝐵𝐵 ∈ V)
4 vex 3418 . . . . . . . 8 𝑓 ∈ V
5 f1stres 7525 . . . . . . . . 9 (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})⟶(𝐵 ∖ ran 𝑓)
6 difexg 5087 . . . . . . . . . . 11 (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
76adantl 474 . . . . . . . . . 10 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝐵 ∖ ran 𝑓) ∈ V)
8 snex 5188 . . . . . . . . . 10 {𝒫 ran 𝐴} ∈ V
9 xpexg 7290 . . . . . . . . . 10 (((𝐵 ∖ ran 𝑓) ∈ V ∧ {𝒫 ran 𝐴} ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}) ∈ V)
107, 8, 9sylancl 577 . . . . . . . . 9 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}) ∈ V)
11 fex2 7453 . . . . . . . . 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)
125, 10, 7, 11mp3an2i 1445 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})) ∈ V)
13 unexg 7289 . . . . . . . 8 ((𝑓 ∈ V ∧ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})) ∈ V) → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
144, 12, 13sylancr 578 . . . . . . 7 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
15 cnvexg 7444 . . . . . . 7 ((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
1614, 15syl 17 . . . . . 6 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
17 rnexg 7429 . . . . . 6 ((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V → ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
1816, 17syl 17 . . . . 5 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
19 simpl 475 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝑓:𝐴1-1𝐵)
20 f1dm 6408 . . . . . . . . . 10 (𝑓:𝐴1-1𝐵 → dom 𝑓 = 𝐴)
214dmex 7431 . . . . . . . . . 10 dom 𝑓 ∈ V
2220, 21syl6eqelr 2875 . . . . . . . . 9 (𝑓:𝐴1-1𝐵𝐴 ∈ V)
2322adantr 473 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐴 ∈ V)
24 simpr 477 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐵 ∈ V)
25 eqid 2778 . . . . . . . . 9 (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) = (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))
2625domss2 8472 . . . . . . . 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)))
2719, 23, 24, 26syl3anc 1351 . . . . . . 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)))
2827simp2d 1123 . . . . . 6 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))))
2927simp1d 1122 . . . . . . 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}))))
30 f1oen3g 8322 . . . . . . 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}))))
3116, 29, 30syl2anc 576 . . . . . 6 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))))
3228, 31jca 504 . . . . 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})))))
33 sseq2 3883 . . . . . 6 (𝑥 = ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) → (𝐴𝑥𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))))
34 breq2 4933 . . . . . 6 (𝑥 = ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) → (𝐵𝑥𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))))
3533, 34anbi12d 621 . . . . 5 (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}))))))
3618, 32, 35elabd 3583 . . . 4 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → ∃𝑥(𝐴𝑥𝐵𝑥))
3736ex 405 . . 3 (𝑓:𝐴1-1𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴𝑥𝐵𝑥)))
3837exlimiv 1889 . 2 (∃𝑓 𝑓:𝐴1-1𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴𝑥𝐵𝑥)))
391, 3, 38sylc 65 1 (𝐴𝐵 → ∃𝑥(𝐴𝑥𝐵𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 387   ∧ w3a 1068   = wceq 1507  ∃wex 1742   ∈ wcel 2050  Vcvv 3415   ∖ cdif 3826   ∪ cun 3827   ⊆ wss 3829  𝒫 cpw 4422  {csn 4441  ∪ cuni 4712   class class class wbr 4929   I cid 5311   × cxp 5405  ◡ccnv 5406  dom cdm 5407  ran crn 5408   ↾ cres 5409   ∘ ccom 5411  ⟶wf 6184  –1-1→wf1 6185  –1-1-onto→wf1o 6187  1st c1st 7499   ≈ cen 8303   ≼ cdom 8304 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-1st 7501  df-2nd 7502  df-en 8307  df-dom 8308 This theorem is referenced by: (None)
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