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Theorem f1cof1blem 45394
Description: Lemma for f1cof1b 45395 and focofob 45398. (Contributed by AV, 18-Sep-2024.)
Hypotheses
Ref Expression
fcores.f (𝜑𝐹:𝐴𝐵)
fcores.e 𝐸 = (ran 𝐹𝐶)
fcores.p 𝑃 = (𝐹𝐶)
fcores.x 𝑋 = (𝐹𝑃)
fcores.g (𝜑𝐺:𝐶𝐷)
fcores.y 𝑌 = (𝐺𝐸)
f1cof1blem.s (𝜑 → ran 𝐹 = 𝐶)
Assertion
Ref Expression
f1cof1blem (𝜑 → ((𝑃 = 𝐴𝐸 = 𝐶) ∧ (𝑋 = 𝐹𝑌 = 𝐺)))

Proof of Theorem f1cof1blem
StepHypRef Expression
1 fcores.p . . . . 5 𝑃 = (𝐹𝐶)
2 f1cof1blem.s . . . . . . 7 (𝜑 → ran 𝐹 = 𝐶)
32eqcomd 2739 . . . . . 6 (𝜑𝐶 = ran 𝐹)
43imaeq2d 6014 . . . . 5 (𝜑 → (𝐹𝐶) = (𝐹 “ ran 𝐹))
51, 4eqtrid 2785 . . . 4 (𝜑𝑃 = (𝐹 “ ran 𝐹))
6 cnvimarndm 6035 . . . . 5 (𝐹 “ ran 𝐹) = dom 𝐹
7 fcores.f . . . . . 6 (𝜑𝐹:𝐴𝐵)
87fdmd 6680 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
96, 8eqtrid 2785 . . . 4 (𝜑 → (𝐹 “ ran 𝐹) = 𝐴)
105, 9eqtrd 2773 . . 3 (𝜑𝑃 = 𝐴)
11 fcores.e . . . 4 𝐸 = (ran 𝐹𝐶)
12 simpr 486 . . . . . . 7 ((𝜑 ∧ ran 𝐹 = 𝐶) → ran 𝐹 = 𝐶)
1312ineq1d 4172 . . . . . 6 ((𝜑 ∧ ran 𝐹 = 𝐶) → (ran 𝐹𝐶) = (𝐶𝐶))
14 inidm 4179 . . . . . 6 (𝐶𝐶) = 𝐶
1513, 14eqtrdi 2789 . . . . 5 ((𝜑 ∧ ran 𝐹 = 𝐶) → (ran 𝐹𝐶) = 𝐶)
162, 15mpdan 686 . . . 4 (𝜑 → (ran 𝐹𝐶) = 𝐶)
1711, 16eqtrid 2785 . . 3 (𝜑𝐸 = 𝐶)
1810, 17jca 513 . 2 (𝜑 → (𝑃 = 𝐴𝐸 = 𝐶))
19 fcores.x . . . 4 𝑋 = (𝐹𝑃)
205, 6eqtrdi 2789 . . . . 5 (𝜑𝑃 = dom 𝐹)
2120reseq2d 5938 . . . 4 (𝜑 → (𝐹𝑃) = (𝐹 ↾ dom 𝐹))
2219, 21eqtrid 2785 . . 3 (𝜑𝑋 = (𝐹 ↾ dom 𝐹))
237freld 6675 . . . 4 (𝜑 → Rel 𝐹)
24 resdm 5983 . . . 4 (Rel 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
2523, 24syl 17 . . 3 (𝜑 → (𝐹 ↾ dom 𝐹) = 𝐹)
2622, 25eqtrd 2773 . 2 (𝜑𝑋 = 𝐹)
27 fcores.y . . . 4 𝑌 = (𝐺𝐸)
28 fcores.g . . . . . . 7 (𝜑𝐺:𝐶𝐷)
2928fdmd 6680 . . . . . 6 (𝜑 → dom 𝐺 = 𝐶)
3017, 29eqtr4d 2776 . . . . 5 (𝜑𝐸 = dom 𝐺)
3130reseq2d 5938 . . . 4 (𝜑 → (𝐺𝐸) = (𝐺 ↾ dom 𝐺))
3227, 31eqtrid 2785 . . 3 (𝜑𝑌 = (𝐺 ↾ dom 𝐺))
3328freld 6675 . . . 4 (𝜑 → Rel 𝐺)
34 resdm 5983 . . . 4 (Rel 𝐺 → (𝐺 ↾ dom 𝐺) = 𝐺)
3533, 34syl 17 . . 3 (𝜑 → (𝐺 ↾ dom 𝐺) = 𝐺)
3632, 35eqtrd 2773 . 2 (𝜑𝑌 = 𝐺)
3718, 26, 36jca32 517 1 (𝜑 → ((𝑃 = 𝐴𝐸 = 𝐶) ∧ (𝑋 = 𝐹𝑌 = 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  cin 3910  ccnv 5633  dom cdm 5634  ran crn 5635  cres 5636  cima 5637  Rel wrel 5639  wf 6493
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-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  f1cof1b  45395
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