Theorem fnfocofob 47542

Description: If the domain of a function 𝐺 equals the range of a function 𝐹, then the composition (𝐺 ∘ 𝐹) is surjective iff 𝐺 is surjective. (Contributed by GL and AV, 29-Sep-2024.)

Assertion

Ref	Expression
fnfocofob	⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))

Proof of Theorem fnfocofob

Step	Hyp	Ref	Expression
1		cnvimarndm 6035	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
2		fndm 6588	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	2	3ad2ant1 1139	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → dom 𝐹 = 𝐴)
4	1, 3	eqtr2id 2787	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (◡𝐹 “ ran 𝐹))
5		imaeq2 6008	. . . . 5 ⊢ (ran 𝐹 = 𝐵 → (◡𝐹 “ ran 𝐹) = (◡𝐹 “ 𝐵))
6	5	3ad2ant3 1141	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → (◡𝐹 “ ran 𝐹) = (◡𝐹 “ 𝐵))
7	4, 6	eqtrd 2774	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (◡𝐹 “ 𝐵))
8		foeq2 6736	. . 3 ⊢ (𝐴 = (◡𝐹 “ 𝐵) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐶 ↔ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐵)–onto→𝐶))
9	7, 8	syl 17	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐶 ↔ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐵)–onto→𝐶))
10		fnfun 6585	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
11		id 22	. . 3 ⊢ (𝐺:𝐵⟶𝐶 → 𝐺:𝐵⟶𝐶)
12		eqimss2 3974	. . 3 ⊢ (ran 𝐹 = 𝐵 → 𝐵 ⊆ ran 𝐹)
13		funfocofob 47541	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐵⟶𝐶 ∧ 𝐵 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐵)–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))
14	10, 11, 12, 13	syl3an 1166	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐵)–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))
15	9, 14	bitrd 280	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))

Syntax hints:   → wi 4   ↔ wb 207   ∧ w3a 1092   = wceq 1547   ⊆ wss 3883  ^◡ccnv 5617  dom cdm 5618  ran crn 5619   “ cima 5621   ∘ ccom 5622  Fun wfun 6479   Fn wfn 6480  ⟶wf 6481  –onto→wfo 6483

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fo 6491  df-fv 6493

This theorem is referenced by:  focofob  47543  f1ocof1ob  47544