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Theorem f1cof1b 47089
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 1137 . . . . . . . . 9 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴𝐵)
2 eqid 2737 . . . . . . . . 9 (ran 𝐹𝐶) = (ran 𝐹𝐶)
3 eqid 2737 . . . . . . . . 9 (𝐹𝐶) = (𝐹𝐶)
4 eqid 2737 . . . . . . . . 9 (𝐹 ↾ (𝐹𝐶)) = (𝐹 ↾ (𝐹𝐶))
5 simp2 1138 . . . . . . . . 9 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐺:𝐶𝐷)
6 eqid 2737 . . . . . . . . 9 (𝐺 ↾ (ran 𝐹𝐶)) = (𝐺 ↾ (ran 𝐹𝐶))
7 simp3 1139 . . . . . . . . 9 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ran 𝐹 = 𝐶)
81, 2, 3, 4, 5, 6, 7f1cof1blem 47086 . . . . . . . 8 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐹 ↾ (𝐹𝐶)) = 𝐹 ∧ (𝐺 ↾ (ran 𝐹𝐶)) = 𝐺)))
9 simpll 767 . . . . . . . 8 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐹 ↾ (𝐹𝐶)) = 𝐹 ∧ (𝐺 ↾ (ran 𝐹𝐶)) = 𝐺)) → (𝐹𝐶) = 𝐴)
10 f1eq2 6800 . . . . . . . 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 223 . . . . . 6 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴1-1𝐷 ↔ (𝐺𝐹):(𝐹𝐶)–1-1𝐷))
13 ancom 460 . . . . . . . . . 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 234 . . . . . . . 8 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)))
1615adantr 480 . . . . . . 7 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):(𝐹𝐶)–1-1𝐷) → (((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)))
171adantr 480 . . . . . . . . 9 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):(𝐹𝐶)–1-1𝐷) → 𝐹:𝐴𝐵)
185adantr 480 . . . . . . . . 9 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):(𝐹𝐶)–1-1𝐷) → 𝐺:𝐶𝐷)
19 simpr 484 . . . . . . . . 9 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):(𝐹𝐶)–1-1𝐷) → (𝐺𝐹):(𝐹𝐶)–1-1𝐷)
2017, 2, 3, 4, 18, 6, 19fcoresf1 47081 . . . . . . . 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 461 . . . . . . 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 771 . . . . . . . . . 10 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → (𝐺 ↾ (ran 𝐹𝐶)) = 𝐺)
23 simpr 484 . . . . . . . . . . 11 (((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) → (ran 𝐹𝐶) = 𝐶)
2423adantr 480 . . . . . . . . . 10 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → (ran 𝐹𝐶) = 𝐶)
25 eqidd 2738 . . . . . . . . . 10 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → 𝐷 = 𝐷)
2622, 24, 25f1eq123d 6840 . . . . . . . . 9 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → ((𝐺 ↾ (ran 𝐹𝐶)):(ran 𝐹𝐶)–1-1𝐷𝐺:𝐶1-1𝐷))
2726biimpd 229 . . . . . . . 8 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → ((𝐺 ↾ (ran 𝐹𝐶)):(ran 𝐹𝐶)–1-1𝐷𝐺:𝐶1-1𝐷))
28 simprr 773 . . . . . . . . . 10 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → (𝐹 ↾ (𝐹𝐶)) = 𝐹)
29 simpll 767 . . . . . . . . . 10 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → (𝐹𝐶) = 𝐴)
3028, 29, 24f1eq123d 6840 . . . . . . . . 9 ((((𝐹𝐶) = 𝐴 ∧ (ran 𝐹𝐶) = 𝐶) ∧ ((𝐺 ↾ (ran 𝐹𝐶)) = 𝐺 ∧ (𝐹 ↾ (𝐹𝐶)) = 𝐹)) → ((𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1→(ran 𝐹𝐶) ↔ 𝐹:𝐴1-1𝐶))
3130biimpd 229 . . . . . . . 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 6749 . . . . . . . . . . . 12 (𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹)
36 ax-1 6 . . . . . . . . . . . 12 (𝐹:𝐴𝐵 → (𝐹:𝐴⟶ran 𝐹𝐹:𝐴𝐵))
3735, 36impbid2 226 . . . . . . . . . . 11 (𝐹:𝐴𝐵 → (𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹))
3837anbi1d 631 . . . . . . . . . 10 (𝐹:𝐴𝐵 → ((𝐹:𝐴𝐵 ∧ Fun 𝐹) ↔ (𝐹:𝐴⟶ran 𝐹 ∧ Fun 𝐹)))
39 df-f1 6566 . . . . . . . . . 10 (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ Fun 𝐹))
40 df-f1 6566 . . . . . . . . . 10 (𝐹:𝐴1-1→ran 𝐹 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ Fun 𝐹))
4138, 39, 403bitr4g 314 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝐹:𝐴1-1𝐵𝐹:𝐴1-1→ran 𝐹))
42413ad2ant1 1134 . . . . . . . 8 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴1-1𝐵𝐹:𝐴1-1→ran 𝐹))
43 f1eq3 6801 . . . . . . . . 9 (ran 𝐹 = 𝐶 → (𝐹:𝐴1-1→ran 𝐹𝐹:𝐴1-1𝐶))
44433ad2ant3 1136 . . . . . . . 8 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴1-1→ran 𝐹𝐹:𝐴1-1𝐶))
4542, 44bitrd 279 . . . . . . 7 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴1-1𝐵𝐹:𝐴1-1𝐶))
4645anbi2d 630 . . . . . 6 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐵) ↔ (𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐶)))
4746adantr 480 . . . . 5 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):𝐴1-1𝐷) → ((𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐵) ↔ (𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐶)))
4834, 47mpbird 257 . . . 4 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):𝐴1-1𝐷) → (𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐵))
4948ancomd 461 . . 3 (((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) ∧ (𝐺𝐹):𝐴1-1𝐷) → (𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷))
5049ex 412 . 2 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴1-1𝐷 → (𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷)))
51 f1cof1 6814 . . . 4 ((𝐺:𝐶1-1𝐷𝐹:𝐴1-1𝐵) → (𝐺𝐹):(𝐹𝐶)–1-1𝐷)
5251ancoms 458 . . 3 ((𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷) → (𝐺𝐹):(𝐹𝐶)–1-1𝐷)
53 imaeq2 6074 . . . . . . . 8 (𝐶 = ran 𝐹 → (𝐹𝐶) = (𝐹 “ ran 𝐹))
54 cnvimarndm 6101 . . . . . . . 8 (𝐹 “ ran 𝐹) = dom 𝐹
5553, 54eqtrdi 2793 . . . . . . 7 (𝐶 = ran 𝐹 → (𝐹𝐶) = dom 𝐹)
5655eqcoms 2745 . . . . . 6 (ran 𝐹 = 𝐶 → (𝐹𝐶) = dom 𝐹)
57563ad2ant3 1136 . . . . 5 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹𝐶) = dom 𝐹)
581fdmd 6746 . . . . 5 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → dom 𝐹 = 𝐴)
5957, 58eqtrd 2777 . . . 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 244 . 2 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐹:𝐴1-1𝐵𝐺:𝐶1-1𝐷) → (𝐺𝐹):𝐴1-1𝐷))
6250, 61impbid 212 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 206  wa 395  w3a 1087   = wceq 1540  cin 3950  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  ccom 5689  Fun wfun 6555  wf 6557  1-1wf1 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-fv 6569
This theorem is referenced by:  f1ocof1ob  47093
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