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Theorem f1cof1b 45299
Description: If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺𝐹) is injective iff 𝐹 and 𝐺 are both injective. (Contributed by GL and AV, 19-Sep-2024.)
Assertion
Ref Expression
f1cof1b ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴1-1𝐷 ↔ (𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷)))

Proof of Theorem f1cof1b
StepHypRef Expression
1 simp1 1136 . . . . . . . . 9 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴𝐵)
2 eqid 2736 . . . . . . . . 9 (ran 𝐹𝐶) = (ran 𝐹𝐶)
3 eqid 2736 . . . . . . . . 9 (𝐹𝐶) = (𝐹𝐶)
4 eqid 2736 . . . . . . . . 9 (𝐹 ↾ (𝐹𝐶)) = (𝐹 ↾ (𝐹𝐶))
5 simp2 1137 . . . . . . . . 9 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐺:𝐶𝐷)
6 eqid 2736 . . . . . . . . 9 (𝐺 ↾ (ran 𝐹𝐶)) = (𝐺 ↾ (ran 𝐹𝐶))
7 simp3 1138 . . . . . . . . 9 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ran 𝐹 = 𝐶)
81, 2, 3, 4, 5, 6, 7f1cof1blem 45298 . . . . . . . 8 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐹 ↾ (𝐹𝐶)) = 𝐹 ∧ (𝐺 ↾ (ran 𝐹𝐶)) = 𝐺)))
9 simpll 765 . . . . . . . 8 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐹 ↾ (𝐹𝐶)) = 𝐹 ∧ (𝐺 ↾ (ran 𝐹𝐶)) = 𝐺)) → (𝐹𝐶) = 𝐴)
10 f1eq2 6734 . . . . . . . 8 ((𝐹𝐶) = 𝐴 → ((𝐺𝐹):(𝐹𝐶)–1-1𝐷 ↔ (𝐺𝐹):𝐴1-1𝐷))
118, 9, 103syl 18 . . . . . . 7 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):(𝐹𝐶)–1-1𝐷 ↔ (𝐺𝐹):𝐴1-1𝐷))
1211bicomd 222 . . . . . 6 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴1-1𝐷 ↔ (𝐺𝐹):(𝐹𝐶)–1-1𝐷))
13 ancom 461 . . . . . . . . . 10 (((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹) ↔ ((𝐹 ↾ (𝐹𝐶)) = 𝐹 ∧ (𝐺 ↾ (ran 𝐹𝐶)) = 𝐺))
1413anbi2i 623 . . . . . . . . 9 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) ↔ (((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐹 ↾ (𝐹𝐶)) = 𝐹 ∧ (𝐺 ↾ (ran 𝐹𝐶)) = 𝐺)))
158, 14sylibr 233 . . . . . . . 8 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)))
1615adantr 481 . . . . . . 7 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):(𝐹𝐶)–1-1𝐷) → (((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)))
171adantr 481 . . . . . . . . 9 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):(𝐹𝐶)–1-1𝐷) → 𝐹:𝐴𝐵)
185adantr 481 . . . . . . . . 9 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):(𝐹𝐶)–1-1𝐷) → 𝐺:𝐶𝐷)
19 simpr 485 . . . . . . . . 9 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):(𝐹𝐶)–1-1𝐷) → (𝐺𝐹):(𝐹𝐶)–1-1𝐷)
2017, 2, 3, 4, 18, 6, 19fcoresf1 45293 . . . . . . . 8 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):(𝐹𝐶)–1-1𝐷) → ((𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1→(ran 𝐹𝐶) ∧ (𝐺 ↾ (ran 𝐹𝐶)):(ran 𝐹𝐶)–1-1𝐷))
2120ancomd 462 . . . . . . 7 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):(𝐹𝐶)–1-1𝐷) → ((𝐺 ↾ (ran 𝐹𝐶)):(ran 𝐹𝐶)–1-1𝐷 ∧ (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1→(ran 𝐹𝐶)))
22 simprl 769 . . . . . . . . . 10 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → (𝐺 ↾ (ran 𝐹𝐶)) = 𝐺)
23 simpr 485 . . . . . . . . . . 11 (((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) → (ran 𝐹𝐶) = 𝐶)
2423adantr 481 . . . . . . . . . 10 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → (ran 𝐹𝐶) = 𝐶)
25 eqidd 2737 . . . . . . . . . 10 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → 𝐷 = 𝐷)
2622, 24, 25f1eq123d 6776 . . . . . . . . 9 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → ((𝐺 ↾ (ran 𝐹𝐶)):(ran 𝐹𝐶)–1-1𝐷𝐺:𝐶1-1𝐷))
2726biimpd 228 . . . . . . . 8 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → ((𝐺 ↾ (ran 𝐹𝐶)):(ran 𝐹𝐶)–1-1𝐷𝐺:𝐶1-1𝐷))
28 simprr 771 . . . . . . . . . 10 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → (𝐹 ↾ (𝐹𝐶)) = 𝐹)
29 simpll 765 . . . . . . . . . 10 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → (𝐹𝐶) = 𝐴)
3028, 29, 24f1eq123d 6776 . . . . . . . . 9 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → ((𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1→(ran 𝐹𝐶) ↔ 𝐹:𝐴1-1𝐶))
3130biimpd 228 . . . . . . . 8 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → ((𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1→(ran 𝐹𝐶) → 𝐹:𝐴1-1𝐶))
3227, 31anim12d 609 . . . . . . 7 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → (((𝐺 ↾ (ran 𝐹𝐶)):(ran 𝐹𝐶)–1-1𝐷 ∧ (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1→(ran 𝐹𝐶)) → (𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐶)))
3316, 21, 32sylc 65 . . . . . 6 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):(𝐹𝐶)–1-1𝐷) → (𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐶))
3412, 33sylbida 592 . . . . 5 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):𝐴1-1𝐷) → (𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐶))
35 ffrn 6682 . . . . . . . . . . . 12 (𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹)
36 ax-1 6 . . . . . . . . . . . 12 (𝐹:𝐴𝐵 → (𝐹:𝐴⟶ran 𝐹𝐹:𝐴𝐵))
3735, 36impbid2 225 . . . . . . . . . . 11 (𝐹:𝐴𝐵 → (𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹))
3837anbi1d 630 . . . . . . . . . 10 (𝐹:𝐴𝐵 → ((𝐹:𝐴𝐵 ∧ Fun 𝐹) ↔ (𝐹:𝐴⟶ran 𝐹 ∧ Fun 𝐹)))
39 df-f1 6501 . . . . . . . . . 10 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
40 df-f1 6501 . . . . . . . . . 10 (𝐹:𝐴1-1→ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ Fun 𝐹))
4138, 39, 403bitr4g 313 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝐹:𝐴1-1𝐵𝐹:𝐴1-1→ran 𝐹))
42413ad2ant1 1133 . . . . . . . 8 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴1-1𝐵𝐹:𝐴1-1→ran 𝐹))
43 f1eq3 6735 . . . . . . . . 9 (ran 𝐹 = 𝐶 → (𝐹:𝐴1-1→ran 𝐹𝐹:𝐴1-1𝐶))
44433ad2ant3 1135 . . . . . . . 8 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴1-1→ran 𝐹𝐹:𝐴1-1𝐶))
4542, 44bitrd 278 . . . . . . 7 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴1-1𝐵𝐹:𝐴1-1𝐶))
4645anbi2d 629 . . . . . 6 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐵) ↔ (𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐶)))
4746adantr 481 . . . . 5 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):𝐴1-1𝐷) → ((𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐵) ↔ (𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐶)))
4834, 47mpbird 256 . . . 4 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):𝐴1-1𝐷) → (𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐵))
4948ancomd 462 . . 3 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):𝐴1-1𝐷) → (𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷))
5049ex 413 . 2 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴1-1𝐷 → (𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷)))
51 f1cof1 6749 . . . 4 ((𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐵) → (𝐺𝐹):(𝐹𝐶)–1-1𝐷)
5251ancoms 459 . . 3 ((𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷) → (𝐺𝐹):(𝐹𝐶)–1-1𝐷)
53 imaeq2 6009 . . . . . . . 8 (𝐶 = ran 𝐹 → (𝐹𝐶) = (𝐹 “ ran 𝐹))
54 cnvimarndm 6034 . . . . . . . 8 (𝐹 “ ran 𝐹) = dom 𝐹
5553, 54eqtrdi 2792 . . . . . . 7 (𝐶 = ran 𝐹 → (𝐹𝐶) = dom 𝐹)
5655eqcoms 2744 . . . . . 6 (ran 𝐹 = 𝐶 → (𝐹𝐶) = dom 𝐹)
57563ad2ant3 1135 . . . . 5 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹𝐶) = dom 𝐹)
581fdmd 6679 . . . . 5 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → dom 𝐹 = 𝐴)
5957, 58eqtrd 2776 . . . 4 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹𝐶) = 𝐴)
6059, 10syl 17 . . 3 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):(𝐹𝐶)–1-1𝐷 ↔ (𝐺𝐹):𝐴1-1𝐷))
6152, 60imbitrid 243 . 2 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷) → (𝐺𝐹):𝐴1-1𝐷))
6250, 61impbid 211 1 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴1-1𝐷 ↔ (𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  cin 3909  ccnv 5632  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  ccom 5637  Fun wfun 6490  wf 6492  1-1wf1 6493
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-fv 6504
This theorem is referenced by:  f1ocof1ob  45303
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