Theorem fompt 7093

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		fof 6772	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
2		fompt.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
3	2	fmpt 7085	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
4	1, 3	sylibr 236	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
5		nfmpt1 5196	. . . . . . 7 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
6	2, 5	nfcxfr 2921	. . . . . 6 ⊢ Ⅎ𝑥𝐹
7	6	foelrnf 7083	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
8		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
9		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑥𝐵
10	6, 8, 9	nffo 6771	. . . . . . 7 ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵
11		simpr 488	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = (𝐹‘𝑥))
12		simpr 488	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
13	4	r19.21bi 3253	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
14	2	fvmpt2 6981	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝑥) = 𝐶)
15	12, 13, 14	syl2anc 593	. . . . . . . . . 10 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
16	15	adantr 484	. . . . . . . . 9 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → (𝐹‘𝑥) = 𝐶)
17	11, 16	eqtrd 2796	. . . . . . . 8 ⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 = (𝐹‘𝑥)) → 𝑦 = 𝐶)
18	17	exp31 423	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑥 ∈ 𝐴 → (𝑦 = (𝐹‘𝑥) → 𝑦 = 𝐶)))
19	10, 18	reximdai 3263	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
20	19	adantr 484	. . . . 5 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
21	7, 20	mpd 15	. . . 4 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
22	21	ralrimiva 3153	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
23	4, 22	jca 519	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))
24	3	birani 507	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → 𝐹:𝐴⟶𝐵)
25		nfv 1933	. . . . 5 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵
26		nfra1 3285	. . . . 5 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶
27	25, 26	nfan 1918	. . . 4 ⊢ Ⅎ𝑦(∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
28		simpll 776	. . . . 5 ⊢ (((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵)
29		rspa 3250	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶 ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
30	29	adantll 724	. . . . 5 ⊢ (((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = 𝐶)
31		nfra1 3285	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵
32		simp3 1150	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝑦 = 𝐶)
33		simpr 488	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
34		rspa 3250	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)
35	33, 34, 14	syl2anc 593	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
36	35	eqcomd 2767	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝐶 = (𝐹‘𝑥))
37	36	3adant3 1144	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝐶 = (𝐹‘𝑥))
38	32, 37	eqtrd 2796	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶) → 𝑦 = (𝐹‘𝑥))
39	38	3exp 1131	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝑦 = 𝐶 → 𝑦 = (𝐹‘𝑥))))
40	31, 39	reximdai 3263	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐶 → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
41	28, 30, 40	sylc 65	. . . 4 ⊢ (((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) ∧ 𝑦 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
42	27, 41	ralrimia 3260	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
43	6	dffo3f 7081	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
44	24, 42, 43	sylanbrc 592	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶) → 𝐹:𝐴–onto→𝐵)
45	23, 44	impbii 211	1 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085   ↦ cmpt 5178  ⟶wf 6511  –onto→wfo 6513  ‘cfv 6515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-fo 6521  df-fv 6523

This theorem is referenced by:  zringfrac  33710  algextdeglem8  33981  disjinfi  45730