mapfienlem3 - Metamath Proof Explorer

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Theorem mapfienlem3 9380

Description: Lemma 3 for mapfien 9381. (Contributed by AV, 3-Jul-2019.) (Revised by AV, 28-Jul-2024.)

**Hypotheses**
Ref	Expression
mapfien.s	⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑_m 𝐴) ∣ 𝑥 finSupp 𝑍}
mapfien.t	⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑_m 𝐶) ∣ 𝑥 finSupp 𝑊}
mapfien.w	⊢ 𝑊 = (𝐺‘𝑍)
mapfien.f	⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
mapfien.g	⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
mapfien.a	⊢ (𝜑 → 𝐴 ∈ 𝑈)
mapfien.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
mapfien.c	⊢ (𝜑 → 𝐶 ∈ 𝑋)
mapfien.d	⊢ (𝜑 → 𝐷 ∈ 𝑌)
mapfien.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

**Assertion**
Ref	Expression
mapfienlem3	⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ 𝑆)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐶 𝑥,𝑔,𝐹 𝑔,𝐺,𝑥 𝜑,𝑔 𝑥,𝐷 𝑆,𝑔 𝑇,𝑔 𝑥,𝑊 𝑥,𝑍 Allowed substitution hints: 𝜑(𝑥) 𝐴(𝑔) 𝐵(𝑔) 𝐶(𝑔) 𝐷(𝑔) 𝑆(𝑥) 𝑇(𝑥) 𝑈(𝑥, 𝑔) 𝑉(𝑥, 𝑔) 𝑊(𝑔) 𝑋(𝑥, 𝑔) 𝑌(𝑥, 𝑔) 𝑍(𝑔)

**Proof of Theorem mapfienlem3**
Step	Hyp	Ref	Expression
1		mapfien.g	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
2		f1ocnv 6831	. . . . . . 7 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → ^◡𝐺:𝐷–1-1-onto→𝐵)
3		f1of 6818	. . . . . . 7 ⊢ (^◡𝐺:𝐷–1-1-onto→𝐵 → ^◡𝐺:𝐷⟶𝐵)
4	1, 2, 3	3syl 19	. . . . . 6 ⊢ (𝜑 → ^◡𝐺:𝐷⟶𝐵)
5	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ^◡𝐺:𝐷⟶𝐵)
6		elrabi 3641	. . . . . . . 8 ⊢ (𝑔 ∈ {𝑥 ∈ (𝐷 ↑_m 𝐶) ∣ 𝑥 finSupp 𝑊} → 𝑔 ∈ (𝐷 ↑_m 𝐶))
7		mapfien.t	. . . . . . . 8 ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑_m 𝐶) ∣ 𝑥 finSupp 𝑊}
8	6, 7	eleq2s 2878	. . . . . . 7 ⊢ (𝑔 ∈ 𝑇 → 𝑔 ∈ (𝐷 ↑_m 𝐶))
9	8	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔 ∈ (𝐷 ↑_m 𝐶))
10		elmapi 8851	. . . . . 6 ⊢ (𝑔 ∈ (𝐷 ↑_m 𝐶) → 𝑔:𝐶⟶𝐷)
11	9, 10	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔:𝐶⟶𝐷)
12	5, 11	fcod 6729	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (^◡𝐺 ∘ 𝑔):𝐶⟶𝐵)
13		mapfien.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
14		f1ocnv 6831	. . . . . 6 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → ^◡𝐹:𝐴–1-1-onto→𝐶)
15		f1of 6818	. . . . . 6 ⊢ (^◡𝐹:𝐴–1-1-onto→𝐶 → ^◡𝐹:𝐴⟶𝐶)
16	13, 14, 15	3syl 19	. . . . 5 ⊢ (𝜑 → ^◡𝐹:𝐴⟶𝐶)
17	16	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ^◡𝐹:𝐴⟶𝐶)
18	12, 17	fcod 6729	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹):𝐴⟶𝐵)
19		mapfien.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
20		mapfien.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
21	19, 20	elmapd 8842	. . . 4 ⊢ (𝜑 → (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ (𝐵 ↑_m 𝐴) ↔ ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹):𝐴⟶𝐵))
22	21	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ (𝐵 ↑_m 𝐴) ↔ ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹):𝐴⟶𝐵))
23	18, 22	mpbird 260	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ (𝐵 ↑_m 𝐴))
24		mapfien.s	. . 3 ⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑_m 𝐴) ∣ 𝑥 finSupp 𝑍}
25		mapfien.w	. . 3 ⊢ 𝑊 = (𝐺‘𝑍)
26		mapfien.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋)
27		mapfien.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑌)
28		mapfien.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
29	24, 7, 25, 13, 1, 20, 19, 26, 27, 28	mapfienlem2 9379	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) finSupp 𝑍)
30		breq1 5106	. . 3 ⊢ (𝑥 = ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) → (𝑥 finSupp 𝑍 ↔ ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) finSupp 𝑍))
31	30, 24	elrab2 3649	. 2 ⊢ (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ 𝑆 ↔ (((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ (𝐵 ↑_m 𝐴) ∧ ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) finSupp 𝑍))
32	23, 29, 31	sylanbrc 595	1 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) ∈ 𝑆)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {crab 3412 class class class wbr 5103 ^◡ccnv 5654 ∘ ccom 5659 ⟶wf 6529 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7414 ↑_m cmap 8829 finSupp cfsupp 9334

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-1o 8458 df-map 8831 df-en 8956 df-dom 8957 df-fin 8959 df-fsupp 9335

This theorem is used by: mapfien 9381

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