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Theorem sbthlem9 8023
Description: Lemma for sbth 8025. (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 8021 . . . . . . 7 ((Fun 𝑓 ∧ Fun 𝑔) → Fun 𝐻)
51, 2, 3sbthlem5 8019 . . . . . . . 8 ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → dom 𝐻 = 𝐴)
65adantrl 751 . . . . . . 7 ((dom 𝑓 = 𝐴 ∧ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴)) → dom 𝐻 = 𝐴)
74, 6anim12i 589 . . . . . 6 (((Fun 𝑓 ∧ Fun 𝑔) ∧ (dom 𝑓 = 𝐴 ∧ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴))) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
87an42s 869 . . . . 5 (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
98adantlr 750 . . . 4 ((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
109adantlr 750 . . 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 8022 . . . 4 ((Fun 𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)
1211adantll 749 . . 3 (((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)
13 simpr 477 . . . . . . 7 ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) → dom 𝑔 = 𝐵)
1413anim1i 591 . . . . . 6 (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) → (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴))
15 df-rn 5090 . . . . . . 7 ran 𝐻 = dom 𝐻
161, 2, 3sbthlem6 8020 . . . . . . 7 ((ran 𝑓𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
1715, 16syl5eqr 2674 . . . . . 6 ((ran 𝑓𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝐻 = 𝐵)
1814, 17sylanr1 683 . . . . 5 ((ran 𝑓𝐵 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝐻 = 𝐵)
1918adantll 749 . . . 4 ((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝐻 = 𝐵)
2019adantlr 750 . . 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 557 . 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 5855 . . . 4 (𝑓:𝐴1-1𝐵 ↔ (𝑓:𝐴𝐵 ∧ Fun 𝑓))
23 df-f 5854 . . . . . 6 (𝑓:𝐴𝐵 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓𝐵))
24 df-fn 5853 . . . . . . 7 (𝑓 Fn 𝐴 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝐴))
2524anbi1i 730 . . . . . 6 ((𝑓 Fn 𝐴 ∧ ran 𝑓𝐵) ↔ ((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵))
2623, 25bitri 264 . . . . 5 (𝑓:𝐴𝐵 ↔ ((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵))
2726anbi1i 730 . . . 4 ((𝑓:𝐴𝐵 ∧ Fun 𝑓) ↔ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓))
2822, 27bitri 264 . . 3 (𝑓:𝐴1-1𝐵 ↔ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓))
29 df-f1 5855 . . . 4 (𝑔:𝐵1-1𝐴 ↔ (𝑔:𝐵𝐴 ∧ Fun 𝑔))
30 df-f 5854 . . . . . 6 (𝑔:𝐵𝐴 ↔ (𝑔 Fn 𝐵 ∧ ran 𝑔𝐴))
31 df-fn 5853 . . . . . . 7 (𝑔 Fn 𝐵 ↔ (Fun 𝑔 ∧ dom 𝑔 = 𝐵))
3231anbi1i 730 . . . . . 6 ((𝑔 Fn 𝐵 ∧ ran 𝑔𝐴) ↔ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴))
3330, 32bitri 264 . . . . 5 (𝑔:𝐵𝐴 ↔ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴))
3433anbi1i 730 . . . 4 ((𝑔:𝐵𝐴 ∧ Fun 𝑔) ↔ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔))
3529, 34bitri 264 . . 3 (𝑔:𝐵1-1𝐴 ↔ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔))
3628, 35anbi12i 732 . 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 6104 . . 3 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻 Fn 𝐴𝐻 Fn 𝐵))
38 df-fn 5853 . . . 4 (𝐻 Fn 𝐴 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
39 df-fn 5853 . . . 4 (𝐻 Fn 𝐵 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐵))
4038, 39anbi12i 732 . . 3 ((𝐻 Fn 𝐴𝐻 Fn 𝐵) ↔ ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun 𝐻 ∧ dom 𝐻 = 𝐵)))
4137, 40bitri 264 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun 𝐻 ∧ dom 𝐻 = 𝐵)))
4221, 36, 413imtr4i 281 1 ((𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  {cab 2612  Vcvv 3191  cdif 3557  cun 3558  wss 3560   cuni 4407  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081  cima 5082  Fun wfun 5844   Fn wfn 5845  wf 5846  1-1wf1 5847  1-1-ontowf1o 5849
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 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857
This theorem is referenced by:  sbthlem10  8024
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