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Theorem sbthlem9 8619
Description: Lemma for sbth 8621. (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 8617 . . . . . . 7 ((Fun 𝑓 ∧ Fun 𝑔) → Fun 𝐻)
51, 2, 3sbthlem5 8615 . . . . . . . 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 615 . . . . . 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 8618 . . . 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 488 . . . . . . 7 ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) → dom 𝑔 = 𝐵)
1413anim1i 617 . . . . . 6 (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) → (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴))
15 df-rn 5530 . . . . . . 7 ran 𝐻 = dom 𝐻
161, 2, 3sbthlem6 8616 . . . . . . 7 ((ran 𝑓𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
1715, 16syl5eqr 2847 . . . . . 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 519 . 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 6329 . . . 4 (𝑓:𝐴1-1𝐵 ↔ (𝑓:𝐴𝐵 ∧ Fun 𝑓))
23 df-f 6328 . . . . . 6 (𝑓:𝐴𝐵 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓𝐵))
24 df-fn 6327 . . . . . . 7 (𝑓 Fn 𝐴 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝐴))
2524anbi1i 626 . . . . . 6 ((𝑓 Fn 𝐴 ∧ ran 𝑓𝐵) ↔ ((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵))
2623, 25bitri 278 . . . . 5 (𝑓:𝐴𝐵 ↔ ((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵))
2726anbi1i 626 . . . 4 ((𝑓:𝐴𝐵 ∧ Fun 𝑓) ↔ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓))
2822, 27bitri 278 . . 3 (𝑓:𝐴1-1𝐵 ↔ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓))
29 df-f1 6329 . . . 4 (𝑔:𝐵1-1𝐴 ↔ (𝑔:𝐵𝐴 ∧ Fun 𝑔))
30 df-f 6328 . . . . . 6 (𝑔:𝐵𝐴 ↔ (𝑔 Fn 𝐵 ∧ ran 𝑔𝐴))
31 df-fn 6327 . . . . . . 7 (𝑔 Fn 𝐵 ↔ (Fun 𝑔 ∧ dom 𝑔 = 𝐵))
3231anbi1i 626 . . . . . 6 ((𝑔 Fn 𝐵 ∧ ran 𝑔𝐴) ↔ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴))
3330, 32bitri 278 . . . . 5 (𝑔:𝐵𝐴 ↔ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴))
3433anbi1i 626 . . . 4 ((𝑔:𝐵𝐴 ∧ Fun 𝑔) ↔ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔))
3529, 34bitri 278 . . 3 (𝑔:𝐵1-1𝐴 ↔ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔))
3628, 35anbi12i 629 . 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 6598 . . 3 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻 Fn 𝐴𝐻 Fn 𝐵))
38 df-fn 6327 . . . 4 (𝐻 Fn 𝐴 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
39 df-fn 6327 . . . 4 (𝐻 Fn 𝐵 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐵))
4038, 39anbi12i 629 . . 3 ((𝐻 Fn 𝐴𝐻 Fn 𝐵) ↔ ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun 𝐻 ∧ dom 𝐻 = 𝐵)))
4137, 40bitri 278 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun 𝐻 ∧ dom 𝐻 = 𝐵)))
4221, 36, 413imtr4i 295 1 ((𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻:𝐴1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  {cab 2776  Vcvv 3441  cdif 3878  cun 3879  wss 3881   cuni 4800  ccnv 5518  dom cdm 5519  ran crn 5520  cres 5521  cima 5522  Fun wfun 6318   Fn wfn 6319  wf 6320  1-1wf1 6321  1-1-ontowf1o 6323
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331
This theorem is referenced by:  sbthlem10  8620
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