Theorem funfocofob 47109

Description: If the domain of a function 𝐺 is a subset of 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
funfocofob	⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))

Proof of Theorem funfocofob

Step	Hyp	Ref	Expression
1		fdmrn 6677	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
2	1	biimpi 216	. . . . . . 7 ⊢ (Fun 𝐹 → 𝐹:dom 𝐹⟶ran 𝐹)
3	2	3ad2ant1 1133	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → 𝐹:dom 𝐹⟶ran 𝐹)
4	3	adantr 480	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵) → 𝐹:dom 𝐹⟶ran 𝐹)
5		eqid 2731	. . . . 5 ⊢ (ran 𝐹 ∩ 𝐴) = (ran 𝐹 ∩ 𝐴)
6		eqid 2731	. . . . 5 ⊢ (◡𝐹 “ 𝐴) = (◡𝐹 “ 𝐴)
7		eqid 2731	. . . . 5 ⊢ (𝐹 ↾ (◡𝐹 “ 𝐴)) = (𝐹 ↾ (◡𝐹 “ 𝐴))
8		simp2 1137	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → 𝐺:𝐴⟶𝐵)
9	8	adantr 480	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵) → 𝐺:𝐴⟶𝐵)
10		eqid 2731	. . . . 5 ⊢ (𝐺 ↾ (ran 𝐹 ∩ 𝐴)) = (𝐺 ↾ (ran 𝐹 ∩ 𝐴))
11		simpr 484	. . . . 5 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵)
12	4, 5, 6, 7, 9, 10, 11	fcoresfo 47102	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵) → (𝐺 ↾ (ran 𝐹 ∩ 𝐴)):(ran 𝐹 ∩ 𝐴)–onto→𝐵)
13	12	ex 412	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵 → (𝐺 ↾ (ran 𝐹 ∩ 𝐴)):(ran 𝐹 ∩ 𝐴)–onto→𝐵))
14		sseqin2 4168	. . . . . . . . 9 ⊢ (𝐴 ⊆ ran 𝐹 ↔ (ran 𝐹 ∩ 𝐴) = 𝐴)
15	14	biimpi 216	. . . . . . . 8 ⊢ (𝐴 ⊆ ran 𝐹 → (ran 𝐹 ∩ 𝐴) = 𝐴)
16	15	3ad2ant3 1135	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (ran 𝐹 ∩ 𝐴) = 𝐴)
17	8	fdmd 6656	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → dom 𝐺 = 𝐴)
18	16, 17	eqtr4d 2769	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (ran 𝐹 ∩ 𝐴) = dom 𝐺)
19	18	reseq2d 5923	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ↾ (ran 𝐹 ∩ 𝐴)) = (𝐺 ↾ dom 𝐺))
20	8	freld 6652	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → Rel 𝐺)
21		resdm 5970	. . . . . 6 ⊢ (Rel 𝐺 → (𝐺 ↾ dom 𝐺) = 𝐺)
22	20, 21	syl 17	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ↾ dom 𝐺) = 𝐺)
23	19, 22	eqtrd 2766	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ↾ (ran 𝐹 ∩ 𝐴)) = 𝐺)
24		eqidd 2732	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → 𝐵 = 𝐵)
25	23, 16, 24	foeq123d 6751	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ↾ (ran 𝐹 ∩ 𝐴)):(ran 𝐹 ∩ 𝐴)–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))
26	13, 25	sylibd 239	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵 → 𝐺:𝐴–onto→𝐵))
27		simpr 484	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → 𝐺:𝐴–onto→𝐵)
28		simpl1 1192	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → Fun 𝐹)
29		simpl3 1194	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → 𝐴 ⊆ ran 𝐹)
30		focofo 6743	. . . 4 ⊢ ((𝐺:𝐴–onto→𝐵 ∧ Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵)
31	27, 28, 29, 30	syl3anc 1373	. . 3 ⊢ (((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) ∧ 𝐺:𝐴–onto→𝐵) → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵)
32	31	ex 412	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → (𝐺:𝐴–onto→𝐵 → (𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵))
33	26, 32	impbid 212	1 ⊢ ((Fun 𝐹 ∧ 𝐺:𝐴⟶𝐵 ∧ 𝐴 ⊆ ran 𝐹) → ((𝐺 ∘ 𝐹):(◡𝐹 “ 𝐴)–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∩ cin 3896   ⊆ wss 3897  ^◡ccnv 5610  dom cdm 5611  ran crn 5612   ↾ cres 5613   “ cima 5614   ∘ ccom 5615  Rel wrel 5616  Fun wfun 6470  ⟶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:  fnfocofob  47110