mapfienlem3 - Metamath Proof Explorer

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Theorem mapfienlem3 9374

Description: Lemma 3 for mapfien 9375. (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 6837	. . . . . . 7 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → ^◡𝐺:𝐷–1-1-onto→𝐵)
3		f1of 6824	. . . . . . 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 3648	. . . . . . . 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 2883	. . . . . . 7 ⊢ (𝑔 ∈ 𝑇 → 𝑔 ∈ (𝐷 ↑_m 𝐶))
9	8	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔 ∈ (𝐷 ↑_m 𝐶))
10		elmapi 8852	. . . . . 6 ⊢ (𝑔 ∈ (𝐷 ↑_m 𝐶) → 𝑔:𝐶⟶𝐷)
11	9, 10	syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → 𝑔:𝐶⟶𝐷)
12	5, 11	fcod 6735	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → (^◡𝐺 ∘ 𝑔):𝐶⟶𝐵)
13		mapfien.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
14		f1ocnv 6837	. . . . . 6 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → ^◡𝐹:𝐴–1-1-onto→𝐶)
15		f1of 6824	. . . . . 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 6735	. . 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 8843	. . . 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 9373	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑇) → ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) finSupp 𝑍)
30		breq1 5114	. . 3 ⊢ (𝑥 = ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) → (𝑥 finSupp 𝑍 ↔ ((^◡𝐺 ∘ 𝑔) ∘ ^◡𝐹) finSupp 𝑍))
31	30, 24	elrab2 3656	. 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 2146 {crab 3418 class class class wbr 5111 ^◡ccnv 5662 ∘ ccom 5667 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7419 ↑_m cmap 8830 finSupp cfsupp 9328

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-1o 8459 df-map 8832 df-en 8950 df-dom 8951 df-fin 8953 df-fsupp 9329

This theorem is used by: mapfien 9375

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