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Theorem f1ocof1ob 44152
Description: If the range of 𝐹 equals the domain of 𝐺, then the composition (𝐺𝐹) is bijective iff 𝐹 and 𝐺 are both bijective. (Contributed by GL and AV, 7-Oct-2024.)
Assertion
Ref Expression
f1ocof1ob ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴1-1-onto𝐷 ↔ (𝐹:𝐴1-1𝐶𝐺:𝐶1-1-onto𝐷)))

Proof of Theorem f1ocof1ob
StepHypRef Expression
1 ffrn 6528 . . . . . . 7 (𝐹:𝐴𝐵𝐹:𝐴⟶ran 𝐹)
213ad2ant1 1134 . . . . . 6 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴⟶ran 𝐹)
3 feq3 6497 . . . . . . 7 (ran 𝐹 = 𝐶 → (𝐹:𝐴⟶ran 𝐹𝐹:𝐴𝐶))
433ad2ant3 1136 . . . . . 6 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (𝐹:𝐴⟶ran 𝐹𝐹:𝐴𝐶))
52, 4mpbid 235 . . . . 5 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → 𝐹:𝐴𝐶)
6 f1cof1b 44148 . . . . 5 ((𝐹:𝐴𝐶𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴1-1𝐷 ↔ (𝐹:𝐴1-1𝐶𝐺:𝐶1-1𝐷)))
75, 6syld3an1 1411 . . . 4 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴1-1𝐷 ↔ (𝐹:𝐴1-1𝐶𝐺:𝐶1-1𝐷)))
8 ffn 6514 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
9 fnfocofob 44150 . . . . 5 ((𝐹 Fn 𝐴𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴onto𝐷𝐺:𝐶onto𝐷))
108, 9syl3an1 1164 . . . 4 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴onto𝐷𝐺:𝐶onto𝐷))
117, 10anbi12d 634 . . 3 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (((𝐺𝐹):𝐴1-1𝐷 ∧ (𝐺𝐹):𝐴onto𝐷) ↔ ((𝐹:𝐴1-1𝐶𝐺:𝐶1-1𝐷) ∧ 𝐺:𝐶onto𝐷)))
12 anass 472 . . 3 (((𝐹:𝐴1-1𝐶𝐺:𝐶1-1𝐷) ∧ 𝐺:𝐶onto𝐷) ↔ (𝐹:𝐴1-1𝐶 ∧ (𝐺:𝐶1-1𝐷𝐺:𝐶onto𝐷)))
1311, 12bitrdi 290 . 2 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → (((𝐺𝐹):𝐴1-1𝐷 ∧ (𝐺𝐹):𝐴onto𝐷) ↔ (𝐹:𝐴1-1𝐶 ∧ (𝐺:𝐶1-1𝐷𝐺:𝐶onto𝐷))))
14 df-f1o 6356 . 2 ((𝐺𝐹):𝐴1-1-onto𝐷 ↔ ((𝐺𝐹):𝐴1-1𝐷 ∧ (𝐺𝐹):𝐴onto𝐷))
15 df-f1o 6356 . . 3 (𝐺:𝐶1-1-onto𝐷 ↔ (𝐺:𝐶1-1𝐷𝐺:𝐶onto𝐷))
1615anbi2i 626 . 2 ((𝐹:𝐴1-1𝐶𝐺:𝐶1-1-onto𝐷) ↔ (𝐹:𝐴1-1𝐶 ∧ (𝐺:𝐶1-1𝐷𝐺:𝐶onto𝐷)))
1713, 14, 163bitr4g 317 1 ((𝐹:𝐴𝐵𝐺:𝐶𝐷 ∧ ran 𝐹 = 𝐶) → ((𝐺𝐹):𝐴1-1-onto𝐷 ↔ (𝐹:𝐴1-1𝐶𝐺:𝐶1-1-onto𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  ran crn 5536  ccom 5539   Fn wfn 6344  wf 6345  1-1wf1 6346  ontowfo 6347  1-1-ontowf1o 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357
This theorem is referenced by:  f1ocof1ob2  44153
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