Theorem fnfocofob 44571

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 5990	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
2		fndm 6536	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	2	3ad2ant1 1132	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → dom 𝐹 = 𝐴)
4	1, 3	eqtr2id 2791	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (◡𝐹 “ ran 𝐹))
5		imaeq2 5965	. . . . 5 ⊢ (ran 𝐹 = 𝐵 → (◡𝐹 “ ran 𝐹) = (◡𝐹 “ 𝐵))
6	5	3ad2ant3 1134	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → (◡𝐹 “ ran 𝐹) = (◡𝐹 “ 𝐵))
7	4, 6	eqtrd 2778	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (◡𝐹 “ 𝐵))
8		foeq2 6685	. . 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 6533	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
11		id 22	. . 3 ⊢ (𝐺:𝐵⟶𝐶 → 𝐺:𝐵⟶𝐶)
12		eqimss2 3978	. . 3 ⊢ (ran 𝐹 = 𝐵 → 𝐵 ⊆ ran 𝐹)
13		funfocofob 44570	. . 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 1159	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐵)–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))
15	9, 14	bitrd 278	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1086   = wceq 1539   ⊆ wss 3887  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   “ cima 5592   ∘ ccom 5593  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  –onto→wfo 6431

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441

This theorem is referenced by:  focofob  44572  f1ocof1ob  44573