Theorem f1mpt 7256

Description: Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
f1mpt.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
f1mpt.2	⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)

Assertion

Ref	Expression
f1mpt	⊢ (𝐹:𝐴–1-1→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)))

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

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

Proof of Theorem f1mpt

Step	Hyp	Ref	Expression
1		f1mpt.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
2		nfmpt1 5255	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
3	1, 2	nfcxfr 2901	. . 3 ⊢ Ⅎ𝑥𝐹
4		nfcv 2903	. . 3 ⊢ Ⅎ𝑦𝐹
5	3, 4	dff13f 7251	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
6	1	fmpt 7106	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
7	6	anbi1i 624	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
8		f1mpt.2	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
9	8	eleq1d 2818	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐶 ∈ 𝐵 ↔ 𝐷 ∈ 𝐵))
10	9	cbvralvw 3234	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝐷 ∈ 𝐵)
11		raaanv 4520	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐴 𝐷 ∈ 𝐵))
12	1	fvmpt2 7006	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝑥) = 𝐶)
13	8, 1	fvmptg 6993	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐹‘𝑦) = 𝐷)
14	12, 13	eqeqan12d 2746	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝐶 = 𝐷))
15	14	an4s 658	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝐶 = 𝐷))
16	15	imbi1d 341	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷 → 𝑥 = 𝑦)))
17	16	ex 413	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷 → 𝑥 = 𝑦))))
18	17	ralimdva 3167	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → ∀𝑦 ∈ 𝐴 (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷 → 𝑥 = 𝑦))))
19		ralbi 3103	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷 → 𝑥 = 𝑦)) → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)))
20	18, 19	syl6 35	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦))))
21	20	ralimia 3080	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)))
22		ralbi 3103	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)))
23	21, 22	syl 17	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)))
24	11, 23	sylbir 234	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐴 𝐷 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)))
25	10, 24	sylan2b 594	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)))
26	25	anidms 567	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)))
27	26	pm5.32i 575	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)))
28	5, 7, 27	3bitr2i 298	1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ↦ cmpt 5230  ⟶wf 6536  –1-1→wf1 6537  ‘cfv 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fv 6548

This theorem is referenced by:  ismon2  17677  isepi2  17684  uspgredg2v  28470  usgredg2v  28473  aciunf1lem  31874  fnpreimac  31883  ply1degltdimlem  32695  ply1degltdim  32696  disjf1  43865