Theorem fompt 41460

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 5166	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
3	1, 2	nfcxfr 2977	. . . . . 6 ⊢ Ⅎ𝑥𝐹
4	3	dffo3f 41445	. . . . 5 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
5	4	simplbi 500	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
6	1	fmpt 6876	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
7	6	bicomi 226	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 ↔ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
8	7	biimpi 218	. . . 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 41454	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
11		nfcv 2979	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
12		nfcv 2979	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
13	3, 11, 12	nffo 6591	. . . . . . 7 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵
14		simpr 487	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = (𝐹‘𝑥))
15		simpr 487	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
16	9	r19.21bi 3210	. . . . . . . . . . . 12 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
17	1	fvmpt2 6781	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝑥) = 𝐶)
18	15, 16, 17	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
19	18	adantr 483	. . . . . . . . . 10 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → (𝐹‘𝑥) = 𝐶)
20	14, 19	eqtrd 2858	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = 𝐶)
21	20	ex 415	. . . . . . . 8 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹‘𝑥) → 𝑦 = 𝐶))
22	21	ex 415	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑥 ∈ 𝐴 → (𝑦 = (𝐹‘𝑥) → 𝑦 = 𝐶)))
23	13, 22	reximdai 3313	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
24	23	adantr 483	. . . . 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 3184	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
27	9, 26	jca 514	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
28	6	biimpi 218	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → 𝐹:𝐴⟶𝐵)
29	28	adantr 483	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → 𝐹:𝐴⟶𝐵)
30		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵
31		nfra1 3221	. . . . . 6 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶
32	30, 31	nfan 1900	. . . . 5 ⊢ Ⅎ𝑦(∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
33		simpll 765	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
34		rspa 3208	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
35	34	adantll 712	. . . . . . 7 ⊢ (((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
36		nfra1 3221	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵
37		simp3 1134	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝑦 = 𝐶)
38		simpr 487	. . . . . . . . . . . . . 14 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
39		rspa 3208	. . . . . . . . . . . . . 14 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
40	38, 39, 17	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
41	40	eqcomd 2829	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 = (𝐹‘𝑥))
42	41	3adant3 1128	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐶 = (𝐹‘𝑥))
43	37, 42	eqtrd 2858	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝑦 = (𝐹‘𝑥))
44	43	3exp 1115	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐶 → 𝑦 = (𝐹‘𝑥))))
45	36, 44	reximdai 3313	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐶 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
46	45	imp 409	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
47	33, 35, 46	syl2anc 586	. . . . . 6 ⊢ (((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
48	47	ex 415	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → (𝑦 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
49	32, 48	ralrimi 3218	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
50	29, 49	jca 514	. . 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 236	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → 𝐹:𝐴–onto→𝐵)
52	27, 51	impbii 211	1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ↦ cmpt 5148  ⟶wf 6353  –onto→wfo 6355  ‘cfv 6357

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fo 6363  df-fv 6365

This theorem is referenced by:  disjinfi  41461