Theorem fnfocofob 47110

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 6027	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
2		fndm 6579	. . . . . 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 2779	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (◡𝐹 “ ran 𝐹))
5		imaeq2 6000	. . . . 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 2766	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐶 ∧ ran 𝐹 = 𝐵) → 𝐴 = (◡𝐹 “ 𝐵))
8		foeq2 6727	. . 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 6576	. . 3 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
11		id 22	. . 3 ⊢ (𝐺:𝐵⟶𝐶 → 𝐺:𝐵⟶𝐶)
12		eqimss2 3989	. . 3 ⊢ (ran 𝐹 = 𝐵 → 𝐵 ⊆ ran 𝐹)
13		funfocofob 47109	. . 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 1541   ⊆ wss 3897  ^◡ccnv 5610  dom cdm 5611  ran crn 5612   “ cima 5614   ∘ ccom 5615  Fun wfun 6470   Fn wfn 6471  ⟶wf 6472  –onto→wfo 6474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-fo 6482  df-fv 6484

This theorem is referenced by:  focofob  47111  f1ocof1ob  47112