Theorem mptcnfimad 8010

Description: The converse of a mapping of subsets to their image of a bijection. (Contributed by AV, 23-Apr-2025.)

Hypotheses

Ref	Expression
mptcnfimad.m	⊢ 𝑀 = (𝑥 ∈ 𝐴 ↦ (𝐹 “ 𝑥))
mptcnfimad.f	⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊)
mptcnfimad.a	⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑉)
mptcnfimad.r	⊢ (𝜑 → ran 𝑀 ⊆ 𝒫 𝑊)
mptcnfimad.v	⊢ (𝜑 → 𝑉 ∈ 𝑈)

Assertion

Ref	Expression
mptcnfimad	⊢ (𝜑 → ◡𝑀 = (𝑦 ∈ ran 𝑀 ↦ (◡𝐹 “ 𝑦)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐹,𝑦   𝑥,𝑀   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑈(𝑥,𝑦)   𝑀(𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem mptcnfimad

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mptcnfimad.m	. . 3 ⊢ 𝑀 = (𝑥 ∈ 𝐴 ↦ (𝐹 “ 𝑥))
2	1	cnveqi 5888	. 2 ⊢ ◡𝑀 = ◡(𝑥 ∈ 𝐴 ↦ (𝐹 “ 𝑥))
3		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
4		mptcnfimad.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊)
5		f1of 6849	. . . . . . . . . . . 12 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉⟶𝑊)
6	4, 5	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑉⟶𝑊)
7		mptcnfimad.v	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ 𝑈)
8	6, 7	fexd 7247	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
9	8	imaexd 7939	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 “ 𝑥) ∈ V)
10	9	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹 “ 𝑥) ∈ V)
11	1, 3, 10	elrnmpt1d 5978	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹 “ 𝑥) ∈ ran 𝑀)
12		f1of1 6848	. . . . . . . . 9 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉–1-1→𝑊)
13	4, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑉–1-1→𝑊)
14		mptcnfimad.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑉)
15		ssel 3989	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝒫 𝑉 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝑉))
16		elpwi 4612	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝒫 𝑉 → 𝑥 ⊆ 𝑉)
17	15, 16	syl6 35	. . . . . . . . . 10 ⊢ (𝐴 ⊆ 𝒫 𝑉 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝑉))
18	14, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝑉))
19	18	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝑉)
20		f1imacnv 6865	. . . . . . . . 9 ⊢ ((𝐹:𝑉–1-1→𝑊 ∧ 𝑥 ⊆ 𝑉) → (◡𝐹 “ (𝐹 “ 𝑥)) = 𝑥)
21	20	eqcomd 2741	. . . . . . . 8 ⊢ ((𝐹:𝑉–1-1→𝑊 ∧ 𝑥 ⊆ 𝑉) → 𝑥 = (◡𝐹 “ (𝐹 “ 𝑥)))
22	13, 19, 21	syl2an2r 685	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = (◡𝐹 “ (𝐹 “ 𝑥)))
23	11, 22	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 “ 𝑥) ∈ ran 𝑀 ∧ 𝑥 = (◡𝐹 “ (𝐹 “ 𝑥))))
24		eleq1 2827	. . . . . . 7 ⊢ (𝑦 = (𝐹 “ 𝑥) → (𝑦 ∈ ran 𝑀 ↔ (𝐹 “ 𝑥) ∈ ran 𝑀))
25		imaeq2 6076	. . . . . . . 8 ⊢ (𝑦 = (𝐹 “ 𝑥) → (◡𝐹 “ 𝑦) = (◡𝐹 “ (𝐹 “ 𝑥)))
26	25	eqeq2d 2746	. . . . . . 7 ⊢ (𝑦 = (𝐹 “ 𝑥) → (𝑥 = (◡𝐹 “ 𝑦) ↔ 𝑥 = (◡𝐹 “ (𝐹 “ 𝑥))))
27	24, 26	anbi12d 632	. . . . . 6 ⊢ (𝑦 = (𝐹 “ 𝑥) → ((𝑦 ∈ ran 𝑀 ∧ 𝑥 = (◡𝐹 “ 𝑦)) ↔ ((𝐹 “ 𝑥) ∈ ran 𝑀 ∧ 𝑥 = (◡𝐹 “ (𝐹 “ 𝑥)))))
28	23, 27	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹 “ 𝑥) → (𝑦 ∈ ran 𝑀 ∧ 𝑥 = (◡𝐹 “ 𝑦))))
29	28	expimpd 453	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹 “ 𝑥)) → (𝑦 ∈ ran 𝑀 ∧ 𝑥 = (◡𝐹 “ 𝑦))))
30	10	ralrimiva 3144	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹 “ 𝑥) ∈ V)
31	1	fnmpt 6709	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 “ 𝑥) ∈ V → 𝑀 Fn 𝐴)
32	30, 31	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 Fn 𝐴)
33		fvelrnb 6969	. . . . . . . . . 10 ⊢ (𝑀 Fn 𝐴 → (𝑦 ∈ ran 𝑀 ↔ ∃𝑥 ∈ 𝐴 (𝑀‘𝑥) = 𝑦))
34	32, 33	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ran 𝑀 ↔ ∃𝑥 ∈ 𝐴 (𝑀‘𝑥) = 𝑦))
35		imaeq2 6076	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐹 “ 𝑥) = (𝐹 “ 𝑧))
36	35	cbvmptv 5261	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹 “ 𝑥)) = (𝑧 ∈ 𝐴 ↦ (𝐹 “ 𝑧))
37	1, 36	eqtri 2763	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (𝑧 ∈ 𝐴 ↦ (𝐹 “ 𝑧))
38	37	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 = (𝑧 ∈ 𝐴 ↦ (𝐹 “ 𝑧)))
39		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 = 𝑥) → 𝑧 = 𝑥)
40	39	imaeq2d 6080	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 = 𝑥) → (𝐹 “ 𝑧) = (𝐹 “ 𝑥))
41	38, 40, 3, 10	fvmptd 7023	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑀‘𝑥) = (𝐹 “ 𝑥))
42	41	eqeq1d 2737	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑀‘𝑥) = 𝑦 ↔ (𝐹 “ 𝑥) = 𝑦))
43	25	eqcoms 2743	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑥) = 𝑦 → (◡𝐹 “ 𝑦) = (◡𝐹 “ (𝐹 “ 𝑥)))
44	43	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹 “ 𝑥) = 𝑦) → (◡𝐹 “ 𝑦) = (◡𝐹 “ (𝐹 “ 𝑥)))
45	13, 19, 20	syl2an2r 685	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ (𝐹 “ 𝑥)) = 𝑥)
46	45, 3	eqeltrd 2839	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐴)
47	46	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹 “ 𝑥) = 𝑦) → (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐴)
48	44, 47	eqeltrd 2839	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹 “ 𝑥) = 𝑦) → (◡𝐹 “ 𝑦) ∈ 𝐴)
49	48	ex 412	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 “ 𝑥) = 𝑦 → (◡𝐹 “ 𝑦) ∈ 𝐴))
50	42, 49	sylbid 240	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑀‘𝑥) = 𝑦 → (◡𝐹 “ 𝑦) ∈ 𝐴))
51	50	rexlimdva 3153	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 (𝑀‘𝑥) = 𝑦 → (◡𝐹 “ 𝑦) ∈ 𝐴))
52	34, 51	sylbid 240	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ran 𝑀 → (◡𝐹 “ 𝑦) ∈ 𝐴))
53	52	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑀) → (◡𝐹 “ 𝑦) ∈ 𝐴)
54		f1ofo 6856	. . . . . . . . . 10 ⊢ (𝐹:𝑉–1-1-onto→𝑊 → 𝐹:𝑉–onto→𝑊)
55	4, 54	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑉–onto→𝑊)
56		mptcnfimad.r	. . . . . . . . . . 11 ⊢ (𝜑 → ran 𝑀 ⊆ 𝒫 𝑊)
57		ssel 3989	. . . . . . . . . . . 12 ⊢ (ran 𝑀 ⊆ 𝒫 𝑊 → (𝑦 ∈ ran 𝑀 → 𝑦 ∈ 𝒫 𝑊))
58		elpwi 4612	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝒫 𝑊 → 𝑦 ⊆ 𝑊)
59	57, 58	syl6 35	. . . . . . . . . . 11 ⊢ (ran 𝑀 ⊆ 𝒫 𝑊 → (𝑦 ∈ ran 𝑀 → 𝑦 ⊆ 𝑊))
60	56, 59	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ∈ ran 𝑀 → 𝑦 ⊆ 𝑊))
61	60	imp 406	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑀) → 𝑦 ⊆ 𝑊)
62		foimacnv 6866	. . . . . . . . 9 ⊢ ((𝐹:𝑉–onto→𝑊 ∧ 𝑦 ⊆ 𝑊) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
63	55, 61, 62	syl2an2r 685	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑀) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
64	63	eqcomd 2741	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑀) → 𝑦 = (𝐹 “ (◡𝐹 “ 𝑦)))
65	53, 64	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑀) → ((◡𝐹 “ 𝑦) ∈ 𝐴 ∧ 𝑦 = (𝐹 “ (◡𝐹 “ 𝑦))))
66		eleq1 2827	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝑥 ∈ 𝐴 ↔ (◡𝐹 “ 𝑦) ∈ 𝐴))
67		imaeq2 6076	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)))
68	67	eqeq2d 2746	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝑦 = (𝐹 “ 𝑥) ↔ 𝑦 = (𝐹 “ (◡𝐹 “ 𝑦))))
69	66, 68	anbi12d 632	. . . . . 6 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹 “ 𝑥)) ↔ ((◡𝐹 “ 𝑦) ∈ 𝐴 ∧ 𝑦 = (𝐹 “ (◡𝐹 “ 𝑦)))))
70	65, 69	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑀) → (𝑥 = (◡𝐹 “ 𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹 “ 𝑥))))
71	70	expimpd 453	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ ran 𝑀 ∧ 𝑥 = (◡𝐹 “ 𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹 “ 𝑥))))
72	29, 71	impbid 212	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹 “ 𝑥)) ↔ (𝑦 ∈ ran 𝑀 ∧ 𝑥 = (◡𝐹 “ 𝑦))))
73	72	mptcnv 6162	. 2 ⊢ (𝜑 → ◡(𝑥 ∈ 𝐴 ↦ (𝐹 “ 𝑥)) = (𝑦 ∈ ran 𝑀 ↦ (◡𝐹 “ 𝑦)))
74	2, 73	eqtrid 2787	1 ⊢ (𝜑 → ◡𝑀 = (𝑦 ∈ ran 𝑀 ↦ (◡𝐹 “ 𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  Vcvv 3478   ⊆ wss 3963  𝒫 cpw 4605   ↦ cmpt 5231  ^◡ccnv 5688  ran crn 5690   “ cima 5692   Fn wfn 6558  ⟶wf 6559  –1-1→wf1 6560  –onto→wfo 6561  –1-1-onto→wf1o 6562  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571

This theorem is referenced by:  uspgrimprop  47811