Theorem f1mpt 7249

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 5204	. . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐶)
3	1, 2	nfcxfr 2925	. . 3 ⊢ Ⅎ𝑥𝐹
4		nfcv 2927	. . 3 ⊢ Ⅎ𝑦𝐹
5	3, 4	dff13f 7243	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
6	1	fmpt 7095	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
7	6	anbi1i 635	. 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 2850	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐶 ∈ 𝐵 ↔ 𝐷 ∈ 𝐵))
10	9	cbvralvw 3243	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 𝐷 ∈ 𝐵)
11		raaanv 4476	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐴 𝐷 ∈ 𝐵))
12	1	fvmpt2 6991	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝑥) = 𝐶)
13	8, 1	fvmptg 6977	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐹‘𝑦) = 𝐷)
14	12, 13	eqeqan12d 2779	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝐶 = 𝐷))
15	14	an4s 672	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ 𝐶 = 𝐷))
16	15	imbi1d 344	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵)) → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷 → 𝑥 = 𝑦)))
17	16	ex 417	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷 → 𝑥 = 𝑦))))
18	17	ralimdva 3177	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → ∀𝑦 ∈ 𝐴 (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (𝐶 = 𝐷 → 𝑥 = 𝑦))))
19		ralbi 3120	. . . . . . . . 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 36	. . . . . . . 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 3099	. . . . . . 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 3120	. . . . . . 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 18	. . . . . 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 238	. . . . 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 605	. . . 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 576	. . 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 584	. 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 302	1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079   ↦ cmpt 5186  ⟶wf 6521  –1-1→wf1 6522  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fv 6533

This theorem is referenced by:  ismon2  17781  isepi2  17788  chninf  18681  uspgredg2v  29483  usgredg2v  29486  aciunf1lem  32919  fnpreimac  32927  rlocf1  33507  zringfrac  33761  selvply1rhmlema  33825  selvply1rhmlem1  33827  ply1degltdimlem  33929  ply1degltdim  33930  assalactf1o  33942  mh-inf3sn  36915  disjf1  45759  upgrimtrls  48526