Theorem mptcnfimad 7991

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 5877	. 2 ⊢ ◡𝑀 = ◡(𝑥 ∈ 𝐴 ↦ (𝐹 “ 𝑥))
3		simpr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
4		mptcnfimad.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊)
5		f1of 6838	. . . . . . . . . . . 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 7239	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ V)
9	8	imaexd 7924	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 “ 𝑥) ∈ V)
10	9	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹 “ 𝑥) ∈ V)
11	1, 3, 10	elrnmpt1d 5964	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹 “ 𝑥) ∈ ran 𝑀)
12		f1of1 6837	. . . . . . . . 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 3970	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ 𝒫 𝑉 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝑉))
16		elpwi 4611	. . . . . . . . . . 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 405	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ 𝑉)
20		f1imacnv 6854	. . . . . . . . 9 ⊢ ((𝐹:𝑉–1-1→𝑊 ∧ 𝑥 ⊆ 𝑉) → (◡𝐹 “ (𝐹 “ 𝑥)) = 𝑥)
21	20	eqcomd 2731	. . . . . . . 8 ⊢ ((𝐹:𝑉–1-1→𝑊 ∧ 𝑥 ⊆ 𝑉) → 𝑥 = (◡𝐹 “ (𝐹 “ 𝑥)))
22	13, 19, 21	syl2an2r 683	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = (◡𝐹 “ (𝐹 “ 𝑥)))
23	11, 22	jca 510	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 “ 𝑥) ∈ ran 𝑀 ∧ 𝑥 = (◡𝐹 “ (𝐹 “ 𝑥))))
24		eleq1 2813	. . . . . . 7 ⊢ (𝑦 = (𝐹 “ 𝑥) → (𝑦 ∈ ran 𝑀 ↔ (𝐹 “ 𝑥) ∈ ran 𝑀))
25		imaeq2 6060	. . . . . . . 8 ⊢ (𝑦 = (𝐹 “ 𝑥) → (◡𝐹 “ 𝑦) = (◡𝐹 “ (𝐹 “ 𝑥)))
26	25	eqeq2d 2736	. . . . . . 7 ⊢ (𝑦 = (𝐹 “ 𝑥) → (𝑥 = (◡𝐹 “ 𝑦) ↔ 𝑥 = (◡𝐹 “ (𝐹 “ 𝑥))))
27	24, 26	anbi12d 630	. . . . . 6 ⊢ (𝑦 = (𝐹 “ 𝑥) → ((𝑦 ∈ ran 𝑀 ∧ 𝑥 = (◡𝐹 “ 𝑦)) ↔ ((𝐹 “ 𝑥) ∈ ran 𝑀 ∧ 𝑥 = (◡𝐹 “ (𝐹 “ 𝑥)))))
28	23, 27	syl5ibrcom 246	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 = (𝐹 “ 𝑥) → (𝑦 ∈ ran 𝑀 ∧ 𝑥 = (◡𝐹 “ 𝑦))))
29	28	expimpd 452	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹 “ 𝑥)) → (𝑦 ∈ ran 𝑀 ∧ 𝑥 = (◡𝐹 “ 𝑦))))
30	10	ralrimiva 3135	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹 “ 𝑥) ∈ V)
31	1	fnmpt 6696	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 (𝐹 “ 𝑥) ∈ V → 𝑀 Fn 𝐴)
32	30, 31	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 Fn 𝐴)
33		fvelrnb 6958	. . . . . . . . . 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 6060	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝐹 “ 𝑥) = (𝐹 “ 𝑧))
36	35	cbvmptv 5262	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐹 “ 𝑥)) = (𝑧 ∈ 𝐴 ↦ (𝐹 “ 𝑧))
37	1, 36	eqtri 2753	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (𝑧 ∈ 𝐴 ↦ (𝐹 “ 𝑧))
38	37	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 = (𝑧 ∈ 𝐴 ↦ (𝐹 “ 𝑧)))
39		simpr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 = 𝑥) → 𝑧 = 𝑥)
40	39	imaeq2d 6064	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑧 = 𝑥) → (𝐹 “ 𝑧) = (𝐹 “ 𝑥))
41	38, 40, 3, 10	fvmptd 7011	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑀‘𝑥) = (𝐹 “ 𝑥))
42	41	eqeq1d 2727	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑀‘𝑥) = 𝑦 ↔ (𝐹 “ 𝑥) = 𝑦))
43	25	eqcoms 2733	. . . . . . . . . . . . . 14 ⊢ ((𝐹 “ 𝑥) = 𝑦 → (◡𝐹 “ 𝑦) = (◡𝐹 “ (𝐹 “ 𝑥)))
44	43	adantl 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹 “ 𝑥) = 𝑦) → (◡𝐹 “ 𝑦) = (◡𝐹 “ (𝐹 “ 𝑥)))
45	13, 19, 20	syl2an2r 683	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ (𝐹 “ 𝑥)) = 𝑥)
46	45, 3	eqeltrd 2825	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐴)
47	46	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹 “ 𝑥) = 𝑦) → (◡𝐹 “ (𝐹 “ 𝑥)) ∈ 𝐴)
48	44, 47	eqeltrd 2825	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹 “ 𝑥) = 𝑦) → (◡𝐹 “ 𝑦) ∈ 𝐴)
49	48	ex 411	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 “ 𝑥) = 𝑦 → (◡𝐹 “ 𝑦) ∈ 𝐴))
50	42, 49	sylbid 239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑀‘𝑥) = 𝑦 → (◡𝐹 “ 𝑦) ∈ 𝐴))
51	50	rexlimdva 3144	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 (𝑀‘𝑥) = 𝑦 → (◡𝐹 “ 𝑦) ∈ 𝐴))
52	34, 51	sylbid 239	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ran 𝑀 → (◡𝐹 “ 𝑦) ∈ 𝐴))
53	52	imp 405	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑀) → (◡𝐹 “ 𝑦) ∈ 𝐴)
54		f1ofo 6845	. . . . . . . . . 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 3970	. . . . . . . . . . . 12 ⊢ (ran 𝑀 ⊆ 𝒫 𝑊 → (𝑦 ∈ ran 𝑀 → 𝑦 ∈ 𝒫 𝑊))
58		elpwi 4611	. . . . . . . . . . . 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 405	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑀) → 𝑦 ⊆ 𝑊)
62		foimacnv 6855	. . . . . . . . 9 ⊢ ((𝐹:𝑉–onto→𝑊 ∧ 𝑦 ⊆ 𝑊) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
63	55, 61, 62	syl2an2r 683	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑀) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
64	63	eqcomd 2731	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑀) → 𝑦 = (𝐹 “ (◡𝐹 “ 𝑦)))
65	53, 64	jca 510	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑀) → ((◡𝐹 “ 𝑦) ∈ 𝐴 ∧ 𝑦 = (𝐹 “ (◡𝐹 “ 𝑦))))
66		eleq1 2813	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝑥 ∈ 𝐴 ↔ (◡𝐹 “ 𝑦) ∈ 𝐴))
67		imaeq2 6060	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)))
68	67	eqeq2d 2736	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝑦 = (𝐹 “ 𝑥) ↔ 𝑦 = (𝐹 “ (◡𝐹 “ 𝑦))))
69	66, 68	anbi12d 630	. . . . . 6 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹 “ 𝑥)) ↔ ((◡𝐹 “ 𝑦) ∈ 𝐴 ∧ 𝑦 = (𝐹 “ (◡𝐹 “ 𝑦)))))
70	65, 69	syl5ibrcom 246	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑀) → (𝑥 = (◡𝐹 “ 𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹 “ 𝑥))))
71	70	expimpd 452	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ ran 𝑀 ∧ 𝑥 = (◡𝐹 “ 𝑦)) → (𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹 “ 𝑥))))
72	29, 71	impbid 211	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = (𝐹 “ 𝑥)) ↔ (𝑦 ∈ ran 𝑀 ∧ 𝑥 = (◡𝐹 “ 𝑦))))
73	72	mptcnv 6146	. 2 ⊢ (𝜑 → ◡(𝑥 ∈ 𝐴 ↦ (𝐹 “ 𝑥)) = (𝑦 ∈ ran 𝑀 ↦ (◡𝐹 “ 𝑦)))
74	2, 73	eqtrid 2777	1 ⊢ (𝜑 → ◡𝑀 = (𝑦 ∈ ran 𝑀 ↦ (◡𝐹 “ 𝑦)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3050  ∃wrex 3059  Vcvv 3461   ⊆ wss 3944  𝒫 cpw 4604   ↦ cmpt 5232  ^◡ccnv 5677  ran crn 5679   “ cima 5681   Fn wfn 6544  ⟶wf 6545  –1-1→wf1 6546  –onto→wfo 6547  –1-1-onto→wf1o 6548  ‘cfv 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557

This theorem is referenced by:  uspgrimprop  47357