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Theorem sbthlem9 7963
 Description: Lemma for sbth 7965. (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 7961 . . . . . . 7 ((Fun 𝑓 ∧ Fun 𝑔) → Fun 𝐻)
51, 2, 3sbthlem5 7959 . . . . . . . 8 ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → dom 𝐻 = 𝐴)
65adantrl 748 . . . . . . 7 ((dom 𝑓 = 𝐴 ∧ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴)) → dom 𝐻 = 𝐴)
74, 6anim12i 588 . . . . . 6 (((Fun 𝑓 ∧ Fun 𝑔) ∧ (dom 𝑓 = 𝐴 ∧ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴))) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
87an42s 866 . . . . 5 (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
98adantlr 747 . . . 4 ((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
109adantlr 747 . . 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 7962 . . . 4 ((Fun 𝑓 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)
1211adantll 746 . . 3 (((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)
13 simpr 476 . . . . . . 7 ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) → dom 𝑔 = 𝐵)
1413anim1i 590 . . . . . 6 (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) → (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴))
15 df-rn 5049 . . . . . . 7 ran 𝐻 = dom 𝐻
161, 2, 3sbthlem6 7960 . . . . . . 7 ((ran 𝑓𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
1715, 16syl5eqr 2658 . . . . . 6 ((ran 𝑓𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝐻 = 𝐵)
1814, 17sylanr1 682 . . . . 5 ((ran 𝑓𝐵 ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝐻 = 𝐵)
1918adantll 746 . . . 4 ((((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝐻 = 𝐵)
2019adantlr 747 . . 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 556 . 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 5809 . . . 4 (𝑓:𝐴1-1𝐵 ↔ (𝑓:𝐴𝐵 ∧ Fun 𝑓))
23 df-f 5808 . . . . . 6 (𝑓:𝐴𝐵 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓𝐵))
24 df-fn 5807 . . . . . . 7 (𝑓 Fn 𝐴 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝐴))
2524anbi1i 727 . . . . . 6 ((𝑓 Fn 𝐴 ∧ ran 𝑓𝐵) ↔ ((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵))
2623, 25bitri 263 . . . . 5 (𝑓:𝐴𝐵 ↔ ((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵))
2726anbi1i 727 . . . 4 ((𝑓:𝐴𝐵 ∧ Fun 𝑓) ↔ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓))
2822, 27bitri 263 . . 3 (𝑓:𝐴1-1𝐵 ↔ (((Fun 𝑓 ∧ dom 𝑓 = 𝐴) ∧ ran 𝑓𝐵) ∧ Fun 𝑓))
29 df-f1 5809 . . . 4 (𝑔:𝐵1-1𝐴 ↔ (𝑔:𝐵𝐴 ∧ Fun 𝑔))
30 df-f 5808 . . . . . 6 (𝑔:𝐵𝐴 ↔ (𝑔 Fn 𝐵 ∧ ran 𝑔𝐴))
31 df-fn 5807 . . . . . . 7 (𝑔 Fn 𝐵 ↔ (Fun 𝑔 ∧ dom 𝑔 = 𝐵))
3231anbi1i 727 . . . . . 6 ((𝑔 Fn 𝐵 ∧ ran 𝑔𝐴) ↔ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴))
3330, 32bitri 263 . . . . 5 (𝑔:𝐵𝐴 ↔ ((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴))
3433anbi1i 727 . . . 4 ((𝑔:𝐵𝐴 ∧ Fun 𝑔) ↔ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔))
3529, 34bitri 263 . . 3 (𝑔:𝐵1-1𝐴 ↔ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔))
3628, 35anbi12i 729 . 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 6058 . . 3 (𝐻:𝐴1-1-onto𝐵 ↔ (𝐻 Fn 𝐴𝐻 Fn 𝐵))
38 df-fn 5807 . . . 4 (𝐻 Fn 𝐴 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐴))
39 df-fn 5807 . . . 4 (𝐻 Fn 𝐵 ↔ (Fun 𝐻 ∧ dom 𝐻 = 𝐵))
4038, 39anbi12i 729 . . 3 ((𝐻 Fn 𝐴𝐻 Fn 𝐵) ↔ ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun 𝐻 ∧ dom 𝐻 = 𝐵)))
4137, 40bitri 263 . 2 (𝐻:𝐴1-1-onto𝐵 ↔ ((Fun 𝐻 ∧ dom 𝐻 = 𝐴) ∧ (Fun 𝐻 ∧ dom 𝐻 = 𝐵)))
4221, 36, 413imtr4i 280 1 ((𝑓:𝐴1-1𝐵𝑔:𝐵1-1𝐴) → 𝐻:𝐴1-1-onto𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∪ cuni 4372  ◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811 This theorem is referenced by:  sbthlem10  7964
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