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Theorem sbthlem9 9091
Description: Lemma for sbth 9093. (Contributed by NM, 28-Mar-1998.)
Hypotheses
Ref Expression
sbthlem.1 𝐴 ∈ V
sbthlem.2 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
sbthlem.3 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
Assertion
Ref Expression
sbthlem9 ((𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻:𝐴1-1-onto𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝑓   𝑥,𝑔   𝑥,𝐻
Allowed substitution hints:   𝐴(𝑓,𝑔)   𝐵(𝑓,𝑔)   𝐷(𝑓,𝑔)   𝐻(𝑓,𝑔)

Proof of Theorem sbthlem9
StepHypRef Expression
1 sbthlem.1 . . . . . . . 8 𝐴 ∈ V
2 sbthlem.2 . . . . . . . 8 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
3 sbthlem.3 . . . . . . . 8 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
41, 2, 3sbthlem7 9089 . . . . . . 7 ((Fun 𝑓 ∧ Fun 𝑔) → Fun 𝐻)
51, 2, 3sbthlem5 9087 . . . . . . . 8 ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → dom 𝐻 = 𝐴)
65adantrl 715 . . . . . . 7 ((dom 𝑓 = 𝐴 ∧ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴)) → dom 𝐻 = 𝐴)
74, 6anim12i 614 . . . . . 6 (((Fun 𝑓 ∧ Fun 𝑔) ∧ (dom 𝑓 = 𝐴 ∧ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴))) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
87an42s 660 . . . . 5 (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
98adantlr 714 . . . 4 ((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
109adantlr 714 . . 3 (((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
111, 2, 3sbthlem8 9090 . . . 4 ((Fun 𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)
1211adantll 713 . . 3 (((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)
13 simpr 486 . . . . . . 7 ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) → dom 𝑔 = 𝐵)
1413anim1i 616 . . . . . 6 (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) → (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴))
15 df-rn 5688 . . . . . . 7 ran 𝐻 = dom 𝐻
161, 2, 3sbthlem6 9088 . . . . . . 7 ((ran 𝑓𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
1715, 16eqtr3id 2787 . . . . . 6 ((ran 𝑓𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝐻 = 𝐵)
1814, 17sylanr1 681 . . . . 5 ((ran 𝑓𝐵 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝐻 = 𝐵)
1918adantll 713 . . . 4 ((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝐻 = 𝐵)
2019adantlr 714 . . 3 (((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝐻 = 𝐵)
2110, 12, 20jca32 517 . 2 (((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun 𝐻 ∧ dom 𝐻 = 𝐵)))
22 df-f1 6549 . . . 4 (𝑓:𝐴1-1𝐵 ↔ (𝑓:𝐴𝐵 ∧ Fun 𝑓))
23 df-f 6548 . . . . . 6 (𝑓:𝐴𝐵 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓𝐵))
24 df-fn 6547 . . . . . . 7 (𝑓 Fn 𝐴 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝐴))
2524anbi1i 625 . . . . . 6 ((𝑓 Fn 𝐴 ∧ ran 𝑓𝐵) ↔ ((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵))
2623, 25bitri 275 . . . . 5 (𝑓:𝐴𝐵 ↔ ((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵))
2726anbi1i 625 . . . 4 ((𝑓:𝐴𝐵 ∧ Fun 𝑓) ↔ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓))
2822, 27bitri 275 . . 3 (𝑓:𝐴1-1𝐵 ↔ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓))
29 df-f1 6549 . . . 4 (𝑔:𝐵1-1𝐴 ↔ (𝑔:𝐵𝐴 ∧ Fun 𝑔))
30 df-f 6548 . . . . . 6 (𝑔:𝐵𝐴 ↔ (𝑔 Fn 𝐵 ∧ ran 𝑔𝐴))
31 df-fn 6547 . . . . . . 7 (𝑔 Fn 𝐵 ↔ (Fun 𝑔 ∧ dom 𝑔 = 𝐵))
3231anbi1i 625 . . . . . 6 ((𝑔 Fn 𝐵 ∧ ran 𝑔𝐴) ↔ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴))
3330, 32bitri 275 . . . . 5 (𝑔:𝐵𝐴 ↔ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴))
3433anbi1i 625 . . . 4 ((𝑔:𝐵𝐴 ∧ Fun 𝑔) ↔ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔))
3529, 34bitri 275 . . 3 (𝑔:𝐵1-1𝐴 ↔ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔))
3628, 35anbi12i 628 . 2 ((𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) ↔ ((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)))
37 dff1o4 6842 . . 3 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻 Fn 𝐴𝐻 Fn 𝐵))
38 df-fn 6547 . . . 4 (𝐻 Fn 𝐴 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
39 df-fn 6547 . . . 4 (𝐻 Fn 𝐵 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐵))
4038, 39anbi12i 628 . . 3 ((𝐻 Fn 𝐴𝐻 Fn 𝐵) ↔ ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun 𝐻 ∧ dom 𝐻 = 𝐵)))
4137, 40bitri 275 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun 𝐻 ∧ dom 𝐻 = 𝐵)))
4221, 36, 413imtr4i 292 1 ((𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  Vcvv 3475  cdif 3946  cun 3947  wss 3949   cuni 4909  ccnv 5676  dom cdm 5677  ran crn 5678  cres 5679  cima 5680  Fun wfun 6538   Fn wfn 6539  wf 6540  1-1wf1 6541  1-1-ontowf1o 6543
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-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  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-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551
This theorem is referenced by:  sbthlem10  9092  sbthfilem  9201
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