Theorem fnfocofob 47634

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 6068	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
2		fndm 6619	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	2	3ad2ant1 1145	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → dom 𝐹 = 𝐴)
4	1, 3	eqtr2id 2809	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (◡𝐹 “ ran 𝐹))
5		imaeq2 6041	. . . . 5 ⊢ (ran 𝐹 = 𝐵 → (◡𝐹 “ ran 𝐹) = (◡𝐹 “ 𝐵))
6	5	3ad2ant3 1147	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → (◡𝐹 “ ran 𝐹) = (◡𝐹 “ 𝐵))
7	4, 6	eqtrd 2796	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (◡𝐹 “ 𝐵))
8		foeq2 6770	. . 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 6616	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
11		id 22	. . 3 ⊢ (𝐺:𝐵⟶𝐶 → 𝐺:𝐵⟶𝐶)
12		eqimss2 3993	. . 3 ⊢ (ran 𝐹 = 𝐵 → 𝐵 ⊆ ran 𝐹)
13		funfocofob 47633	. . 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 1172	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐵)–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))
15	9, 14	bitrd 281	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1097   = wceq 1559   ⊆ wss 3902  ^◡ccnv 5642  dom cdm 5643  ran crn 5644   “ cima 5646   ∘ ccom 5647  Fun wfun 6510   Fn wfn 6511  ⟶wf 6512  –onto→wfo 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-fo 6522  df-fv 6524

This theorem is referenced by:  focofob  47635  f1ocof1ob  47636