Theorem fompt 42958

Description: Express being onto for a mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
fompt.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)

Assertion

Ref	Expression
fompt	⊢ (𝐹:𝐴–onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑦,𝐶   𝑦,𝐹

Allowed substitution hints:   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem fompt

Step	Hyp	Ref	Expression
1		fompt.1	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
2		nfmpt1 5195	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
3	1, 2	nfcxfr 2903	. . . . . 6 ⊢ Ⅎ𝑥𝐹
4	3	dffo3f 42945	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
5	4	simplbi 498	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
6	1	fmpt 7023	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
7	6	bicomi 223	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
8	7	biimpi 215	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
9	5, 8	syl 17	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
10	3	foelrnf 42952	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
11		nfcv 2905	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
12		nfcv 2905	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
13	3, 11, 12	nffo 6724	. . . . . . 7 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵
14		simpr 485	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = (𝐹‘𝑥))
15		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
16	9	r19.21bi 3231	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
17	1	fvmpt2 6925	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝑥) = 𝐶)
18	15, 16, 17	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
19	18	adantr 481	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → (𝐹‘𝑥) = 𝐶)
20	14, 19	eqtrd 2777	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = 𝐶)
21	20	ex 413	. . . . . . . 8 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) → 𝑦 = 𝐶))
22	21	ex 413	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑥 ∈ 𝐴 → (𝑦 = (𝐹‘𝑥) → 𝑦 = 𝐶)))
23	13, 22	reximdai 3241	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
24	23	adantr 481	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
25	10, 24	mpd 15	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
26	25	ralrimiva 3140	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
27	9, 26	jca 512	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
28	6	biimpi 215	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → 𝐹:𝐴⟶𝐵)
29	28	adantr 481	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → 𝐹:𝐴⟶𝐵)
30		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵
31		nfra1 3264	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶
32	30, 31	nfan 1901	. . . . 5 ⊢ Ⅎ𝑦(∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
33		simpll 764	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
34		rspa 3228	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
35	34	adantll 711	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
36		nfra1 3264	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵
37		simp3 1137	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝑦 = 𝐶)
38		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
39		rspa 3228	. . . . . . . . . . . . . 14 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
40	38, 39, 17	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
41	40	eqcomd 2743	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 = (𝐹‘𝑥))
42	41	3adant3 1131	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐶 = (𝐹‘𝑥))
43	37, 42	eqtrd 2777	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝑦 = (𝐹‘𝑥))
44	43	3exp 1118	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐶 → 𝑦 = (𝐹‘𝑥))))
45	36, 44	reximdai 3241	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐶 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
46	45	imp 407	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
47	33, 35, 46	syl2anc 584	. . . . . 6 ⊢ (((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
48	47	ex 413	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
49	32, 48	ralrimi 3237	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
50	29, 49	jca 512	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
51	50, 4	sylibr 233	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → 𝐹:𝐴–onto→𝐵)
52	27, 51	impbii 208	1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3062  ∃wrex 3071   ↦ cmpt 5170  ⟶wf 6461  –onto→wfo 6463  ‘cfv 6465

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-fo 6471  df-fv 6473

This theorem is referenced by:  disjinfi  42959