Theorem f1cof1blem 46084

Description: Lemma for f1cof1b 46085 and focofob 46088. (Contributed by AV, 18-Sep-2024.)

Hypotheses

Ref	Expression
fcores.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
fcores.e	⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
fcores.p	⊢ 𝑃 = (◡𝐹 “ 𝐶)
fcores.x	⊢ 𝑋 = (𝐹 ↾ 𝑃)
fcores.g	⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
fcores.y	⊢ 𝑌 = (𝐺 ↾ 𝐸)
f1cof1blem.s	⊢ (𝜑 → ran 𝐹 = 𝐶)

Assertion

Ref	Expression
f1cof1blem	⊢ (𝜑 → ((𝑃 = 𝐴 ∧ 𝐸 = 𝐶) ∧ (𝑋 = 𝐹 ∧ 𝑌 = 𝐺)))

Proof of Theorem f1cof1blem

Step	Hyp	Ref	Expression
1		fcores.p	. . . . 5 ⊢ 𝑃 = (◡𝐹 “ 𝐶)
2		f1cof1blem.s	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = 𝐶)
3	2	eqcomd 2737	. . . . . 6 ⊢ (𝜑 → 𝐶 = ran 𝐹)
4	3	imaeq2d 6060	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝐶) = (◡𝐹 “ ran 𝐹))
5	1, 4	eqtrid 2783	. . . 4 ⊢ (𝜑 → 𝑃 = (◡𝐹 “ ran 𝐹))
6		cnvimarndm 6082	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
7		fcores.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
8	7	fdmd 6729	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
9	6, 8	eqtrid 2783	. . . 4 ⊢ (𝜑 → (◡𝐹 “ ran 𝐹) = 𝐴)
10	5, 9	eqtrd 2771	. . 3 ⊢ (𝜑 → 𝑃 = 𝐴)
11		fcores.e	. . . 4 ⊢ 𝐸 = (ran 𝐹 ∩ 𝐶)
12		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ ran 𝐹 = 𝐶) → ran 𝐹 = 𝐶)
13	12	ineq1d 4212	. . . . . 6 ⊢ ((𝜑 ∧ ran 𝐹 = 𝐶) → (ran 𝐹 ∩ 𝐶) = (𝐶 ∩ 𝐶))
14		inidm 4219	. . . . . 6 ⊢ (𝐶 ∩ 𝐶) = 𝐶
15	13, 14	eqtrdi 2787	. . . . 5 ⊢ ((𝜑 ∧ ran 𝐹 = 𝐶) → (ran 𝐹 ∩ 𝐶) = 𝐶)
16	2, 15	mpdan 684	. . . 4 ⊢ (𝜑 → (ran 𝐹 ∩ 𝐶) = 𝐶)
17	11, 16	eqtrid 2783	. . 3 ⊢ (𝜑 → 𝐸 = 𝐶)
18	10, 17	jca 511	. 2 ⊢ (𝜑 → (𝑃 = 𝐴 ∧ 𝐸 = 𝐶))
19		fcores.x	. . . 4 ⊢ 𝑋 = (𝐹 ↾ 𝑃)
20	5, 6	eqtrdi 2787	. . . . 5 ⊢ (𝜑 → 𝑃 = dom 𝐹)
21	20	reseq2d 5982	. . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝑃) = (𝐹 ↾ dom 𝐹))
22	19, 21	eqtrid 2783	. . 3 ⊢ (𝜑 → 𝑋 = (𝐹 ↾ dom 𝐹))
23	7	freld 6724	. . . 4 ⊢ (𝜑 → Rel 𝐹)
24		resdm 6027	. . . 4 ⊢ (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
25	23, 24	syl 17	. . 3 ⊢ (𝜑 → (𝐹 ↾ dom 𝐹) = 𝐹)
26	22, 25	eqtrd 2771	. 2 ⊢ (𝜑 → 𝑋 = 𝐹)
27		fcores.y	. . . 4 ⊢ 𝑌 = (𝐺 ↾ 𝐸)
28		fcores.g	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐶⟶𝐷)
29	28	fdmd 6729	. . . . . 6 ⊢ (𝜑 → dom 𝐺 = 𝐶)
30	17, 29	eqtr4d 2774	. . . . 5 ⊢ (𝜑 → 𝐸 = dom 𝐺)
31	30	reseq2d 5982	. . . 4 ⊢ (𝜑 → (𝐺 ↾ 𝐸) = (𝐺 ↾ dom 𝐺))
32	27, 31	eqtrid 2783	. . 3 ⊢ (𝜑 → 𝑌 = (𝐺 ↾ dom 𝐺))
33	28	freld 6724	. . . 4 ⊢ (𝜑 → Rel 𝐺)
34		resdm 6027	. . . 4 ⊢ (Rel 𝐺 → (𝐺 ↾ dom 𝐺) = 𝐺)
35	33, 34	syl 17	. . 3 ⊢ (𝜑 → (𝐺 ↾ dom 𝐺) = 𝐺)
36	32, 35	eqtrd 2771	. 2 ⊢ (𝜑 → 𝑌 = 𝐺)
37	18, 26, 36	jca32 515	1 ⊢ (𝜑 → ((𝑃 = 𝐴 ∧ 𝐸 = 𝐶) ∧ (𝑋 = 𝐹 ∧ 𝑌 = 𝐺)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∩ cin 3948  ^◡ccnv 5676  dom cdm 5677  ran crn 5678   ↾ cres 5679   “ cima 5680  Rel wrel 5682  ⟶wf 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-f 6548

This theorem is referenced by:  f1cof1b  46085