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Theorem domssex 9137
Description: Weakening of domssex2 9136 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 (𝐴 β‰Ό 𝐡 β†’ βˆƒπ‘₯(𝐴 βŠ† π‘₯ ∧ 𝐡 β‰ˆ π‘₯))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡
Proof of Theorem domssex
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 brdomi 8953 . 2 (𝐴 β‰Ό 𝐡 β†’ βˆƒπ‘“ 𝑓:𝐴–1-1→𝐡)
2 reldom 8944 . . 3 Rel β‰Ό
32brrelex2i 5733 . 2 (𝐴 β‰Ό 𝐡 β†’ 𝐡 ∈ V)
4 vex 3478 . . . . . . . 8 𝑓 ∈ V
5 f1stres 7998 . . . . . . . . 9 (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})):((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})⟢(𝐡 βˆ– ran 𝑓)
6 difexg 5327 . . . . . . . . . . 11 (𝐡 ∈ V β†’ (𝐡 βˆ– ran 𝑓) ∈ V)
76adantl 482 . . . . . . . . . 10 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ (𝐡 βˆ– ran 𝑓) ∈ V)
8 snex 5431 . . . . . . . . . 10 {𝒫 βˆͺ ran 𝐴} ∈ V
9 xpexg 7736 . . . . . . . . . 10 (((𝐡 βˆ– ran 𝑓) ∈ V ∧ {𝒫 βˆͺ ran 𝐴} ∈ V) β†’ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}) ∈ V)
107, 8, 9sylancl 586 . . . . . . . . 9 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}) ∈ V)
11 fex2 7923 . . . . . . . . 9 (((1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})):((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})⟢(𝐡 βˆ– ran 𝑓) ∧ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}) ∈ V ∧ (𝐡 βˆ– ran 𝑓) ∈ V) β†’ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})) ∈ V)
125, 10, 7, 11mp3an2i 1466 . . . . . . . 8 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})) ∈ V)
13 unexg 7735 . . . . . . . 8 ((𝑓 ∈ V ∧ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})) ∈ V) β†’ (𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
144, 12, 13sylancr 587 . . . . . . 7 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ (𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
15 cnvexg 7914 . . . . . . 7 ((𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V β†’ β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
1614, 15syl 17 . . . . . 6 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
17 rnexg 7894 . . . . . 6 (β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V β†’ ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
1816, 17syl 17 . . . . 5 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
19 simpl 483 . . . . . . . 8 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ 𝑓:𝐴–1-1→𝐡)
20 f1dm 6791 . . . . . . . . . 10 (𝑓:𝐴–1-1→𝐡 β†’ dom 𝑓 = 𝐴)
214dmex 7901 . . . . . . . . . 10 dom 𝑓 ∈ V
2220, 21eqeltrrdi 2842 . . . . . . . . 9 (𝑓:𝐴–1-1→𝐡 β†’ 𝐴 ∈ V)
2322adantr 481 . . . . . . . 8 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ 𝐴 ∈ V)
24 simpr 485 . . . . . . . 8 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ 𝐡 ∈ V)
25 eqid 2732 . . . . . . . . 9 β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) = β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})))
2625domss2 9135 . . . . . . . 8 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐴 ∈ V ∧ 𝐡 ∈ V) β†’ (β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ 𝐴 βŠ† ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ (β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∘ 𝑓) = ( I β†Ύ 𝐴)))
2719, 23, 24, 26syl3anc 1371 . . . . . . 7 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ (β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ 𝐴 βŠ† ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ (β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∘ 𝑓) = ( I β†Ύ 𝐴)))
2827simp2d 1143 . . . . . 6 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ 𝐴 βŠ† ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))))
2927simp1d 1142 . . . . . . 7 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))))
30 f1oen3g 8961 . . . . . . 7 ((β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V ∧ β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})))) β†’ 𝐡 β‰ˆ ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))))
3116, 29, 30syl2anc 584 . . . . . 6 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ 𝐡 β‰ˆ ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))))
3228, 31jca 512 . . . . 5 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ (𝐴 βŠ† ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ 𝐡 β‰ˆ ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})))))
33 sseq2 4008 . . . . . 6 (π‘₯ = ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) β†’ (𝐴 βŠ† π‘₯ ↔ 𝐴 βŠ† ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})))))
34 breq2 5152 . . . . . 6 (π‘₯ = ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) β†’ (𝐡 β‰ˆ π‘₯ ↔ 𝐡 β‰ˆ ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})))))
3533, 34anbi12d 631 . . . . 5 (π‘₯ = ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) β†’ ((𝐴 βŠ† π‘₯ ∧ 𝐡 β‰ˆ π‘₯) ↔ (𝐴 βŠ† ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ 𝐡 β‰ˆ ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))))))
3618, 32, 35spcedv 3588 . . . 4 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ βˆƒπ‘₯(𝐴 βŠ† π‘₯ ∧ 𝐡 β‰ˆ π‘₯))
3736ex 413 . . 3 (𝑓:𝐴–1-1→𝐡 β†’ (𝐡 ∈ V β†’ βˆƒπ‘₯(𝐴 βŠ† π‘₯ ∧ 𝐡 β‰ˆ π‘₯)))
3837exlimiv 1933 . 2 (βˆƒπ‘“ 𝑓:𝐴–1-1→𝐡 β†’ (𝐡 ∈ V β†’ βˆƒπ‘₯(𝐴 βŠ† π‘₯ ∧ 𝐡 β‰ˆ π‘₯)))
391, 3, 38sylc 65 1 (𝐴 β‰Ό 𝐡 β†’ βˆƒπ‘₯(𝐴 βŠ† π‘₯ ∧ 𝐡 β‰ˆ π‘₯))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  Vcvv 3474   βˆ– cdif 3945   βˆͺ cun 3946   βŠ† wss 3948  π’« cpw 4602  {csn 4628  βˆͺ cuni 4908   class class class wbr 5148   I cid 5573   Γ— cxp 5674  β—‘ccnv 5675  dom cdm 5676  ran crn 5677   β†Ύ cres 5678   ∘ ccom 5680  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  1st c1st 7972   β‰ˆ cen 8935   β‰Ό cdom 8936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1st 7974  df-2nd 7975  df-en 8939  df-dom 8940
This theorem is referenced by: (None)
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