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Theorem domssex 9138
Description: Weakening of domssex2 9137 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 8954 . 2 (𝐴 β‰Ό 𝐡 β†’ βˆƒπ‘“ 𝑓:𝐴–1-1→𝐡)
2 reldom 8945 . . 3 Rel β‰Ό
32brrelex2i 5734 . 2 (𝐴 β‰Ό 𝐡 β†’ 𝐡 ∈ V)
4 vex 3479 . . . . . . . 8 𝑓 ∈ V
5 f1stres 7999 . . . . . . . . 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 5328 . . . . . . . . . . 11 (𝐡 ∈ V β†’ (𝐡 βˆ– ran 𝑓) ∈ V)
76adantl 483 . . . . . . . . . 10 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ (𝐡 βˆ– ran 𝑓) ∈ V)
8 snex 5432 . . . . . . . . . 10 {𝒫 βˆͺ ran 𝐴} ∈ V
9 xpexg 7737 . . . . . . . . . 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 587 . . . . . . . . 9 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}) ∈ V)
11 fex2 7924 . . . . . . . . 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 1467 . . . . . . . 8 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})) ∈ V)
13 unexg 7736 . . . . . . . 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 588 . . . . . . 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 7915 . . . . . . 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 7895 . . . . . 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 484 . . . . . . . 8 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ 𝑓:𝐴–1-1→𝐡)
20 f1dm 6792 . . . . . . . . . 10 (𝑓:𝐴–1-1→𝐡 β†’ dom 𝑓 = 𝐴)
214dmex 7902 . . . . . . . . . 10 dom 𝑓 ∈ V
2220, 21eqeltrrdi 2843 . . . . . . . . 9 (𝑓:𝐴–1-1→𝐡 β†’ 𝐴 ∈ V)
2322adantr 482 . . . . . . . 8 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ 𝐴 ∈ V)
24 simpr 486 . . . . . . . 8 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ 𝐡 ∈ V)
25 eqid 2733 . . . . . . . . 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 9136 . . . . . . . 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 1372 . . . . . . 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 1144 . . . . . 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 1143 . . . . . . 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 8962 . . . . . . 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 585 . . . . . 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 513 . . . . 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 4009 . . . . . 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 5153 . . . . . 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 632 . . . . 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 3589 . . . 4 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ βˆƒπ‘₯(𝐴 βŠ† π‘₯ ∧ 𝐡 β‰ˆ π‘₯))
3736ex 414 . . 3 (𝑓:𝐴–1-1→𝐡 β†’ (𝐡 ∈ V β†’ βˆƒπ‘₯(𝐴 βŠ† π‘₯ ∧ 𝐡 β‰ˆ π‘₯)))
3837exlimiv 1934 . 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 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3475   βˆ– cdif 3946   βˆͺ cun 3947   βŠ† wss 3949  π’« cpw 4603  {csn 4629  βˆͺ cuni 4909   class class class wbr 5149   I cid 5574   Γ— cxp 5675  β—‘ccnv 5676  dom cdm 5677  ran crn 5678   β†Ύ cres 5679   ∘ ccom 5681  βŸΆwf 6540  β€“1-1β†’wf1 6541  β€“1-1-ontoβ†’wf1o 6543  1st c1st 7973   β‰ˆ cen 8936   β‰Ό cdom 8937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-1st 7975  df-2nd 7976  df-en 8940  df-dom 8941
This theorem is referenced by: (None)
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