Theorem fnfocofob 47053

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 6043	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
2		fndm 6603	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	2	3ad2ant1 1133	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → dom 𝐹 = 𝐴)
4	1, 3	eqtr2id 2777	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (◡𝐹 “ ran 𝐹))
5		imaeq2 6016	. . . . 5 ⊢ (ran 𝐹 = 𝐵 → (◡𝐹 “ ran 𝐹) = (◡𝐹 “ 𝐵))
6	5	3ad2ant3 1135	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → (◡𝐹 “ ran 𝐹) = (◡𝐹 “ 𝐵))
7	4, 6	eqtrd 2764	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (◡𝐹 “ 𝐵))
8		foeq2 6751	. . 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 6600	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
11		id 22	. . 3 ⊢ (𝐺:𝐵⟶𝐶 → 𝐺:𝐵⟶𝐶)
12		eqimss2 4003	. . 3 ⊢ (ran 𝐹 = 𝐵 → 𝐵 ⊆ ran 𝐹)
13		funfocofob 47052	. . 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 1160	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐵)–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))
15	9, 14	bitrd 279	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → ((𝐺 ∘ 𝐹):𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1540   ⊆ wss 3911  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   “ cima 5634   ∘ ccom 5635  Fun wfun 6493   Fn wfn 6494  ⟶wf 6495  –onto→wfo 6497

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fo 6505  df-fv 6507

This theorem is referenced by:  focofob  47054  f1ocof1ob  47055