Theorem mapfien2 9438

Description: Equinumerousity relation for sets of finitely supported functions. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)

Hypotheses

Ref	Expression
mapfien2.s	⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 0 }
mapfien2.t	⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}
mapfien2.ac	⊢ (𝜑 → 𝐴 ≈ 𝐶)
mapfien2.bd	⊢ (𝜑 → 𝐵 ≈ 𝐷)
mapfien2.z	⊢ (𝜑 → 0 ∈ 𝐵)
mapfien2.w	⊢ (𝜑 → 𝑊 ∈ 𝐷)

Assertion

Ref	Expression
mapfien2	⊢ (𝜑 → 𝑆 ≈ 𝑇)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝑥, 0   𝑥,𝑊

Allowed substitution hints:   𝜑(𝑥)   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem mapfien2

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapfien2.z	. . 3 ⊢ (𝜑 → 0 ∈ 𝐵)
2		mapfien2.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝐷)
3		mapfien2.bd	. . 3 ⊢ (𝜑 → 𝐵 ≈ 𝐷)
4		enfixsn 9143	. . 3 ⊢ (( 0 ∈ 𝐵 ∧ 𝑊 ∈ 𝐷 ∧ 𝐵 ≈ 𝐷) → ∃𝑦(𝑦:𝐵⟶1-1onto𝐷 ∧ (𝑦‘ 0 ) = 𝑊))
5	1, 2, 3, 4	syl3anc 1399	. 2 ⊢ (𝜑 → ∃𝑦(𝑦:𝐵⟶1-1onto𝐷 ∧ (𝑦‘ 0 ) = 𝑊))
6		mapfien2.ac	. . . . 5 ⊢ (𝜑 → 𝐴 ≈ 𝐶)
7		bren 9021	. . . . 5 ⊢ (𝐴 ≈ 𝐶 ↔ ∃𝑧 𝑧:𝐴⟶1-1onto𝐶)
8	6, 7	sylib 221	. . . 4 ⊢ (𝜑 → ∃𝑧 𝑧:𝐴⟶1-1onto𝐶)
9		mapfien2.s	. . . . . . . . . 10 ⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 0 }
10		eqid 2763	. . . . . . . . . 10 ⊢ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )}
11		eqid 2763	. . . . . . . . . 10 ⊢ (𝑦‘ 0 ) = (𝑦‘ 0 )
12		f1ocnv 6867	. . . . . . . . . . 11 ⊢ (𝑧:𝐴⟶1-1onto𝐶 → ◡𝑧:𝐶⟶1-1onto𝐴)
13	12	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷) → ◡𝑧:𝐶⟶1-1onto𝐴)
14		simp3 1156	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷) → 𝑦:𝐵⟶1-1onto𝐷)
15	6	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷) → 𝐴 ≈ 𝐶)
16		relen 9016	. . . . . . . . . . . 12 ⊢ Rel ≈
17	16	brrelex1i 5714	. . . . . . . . . . 11 ⊢ (𝐴 ≈ 𝐶 → 𝐴 ∈ V)
18	15, 17	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷) → 𝐴 ∈ V)
19	3	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷) → 𝐵 ≈ 𝐷)
20	16	brrelex1i 5714	. . . . . . . . . . 11 ⊢ (𝐵 ≈ 𝐷 → 𝐵 ∈ V)
21	19, 20	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷) → 𝐵 ∈ V)
22	16	brrelex2i 5715	. . . . . . . . . . 11 ⊢ (𝐴 ≈ 𝐶 → 𝐶 ∈ V)
23	15, 22	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷) → 𝐶 ∈ V)
24	16	brrelex2i 5715	. . . . . . . . . . 11 ⊢ (𝐵 ≈ 𝐷 → 𝐷 ∈ V)
25	19, 24	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷) → 𝐷 ∈ V)
26	1	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷) → 0 ∈ 𝐵)
27	9, 10, 11, 13, 14, 18, 21, 23, 25, 26	mapfien 9437	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷) → (𝑤 ∈ 𝑆 ↦ (𝑦 ∘ (𝑤 ∘ ◡𝑧))):𝑆⟶1-1onto{𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )})
28		ovex 7484	. . . . . . . . . . 11 ⊢ (𝐵 ↑m 𝐴) ∈ V
29	9, 28	rabex2 5309	. . . . . . . . . 10 ⊢ 𝑆 ∈ V
30	29	f1oen 9037	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝑆 ↦ (𝑦 ∘ (𝑤 ∘ ◡𝑧))):𝑆⟶1-1onto{𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} → 𝑆 ≈ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )})
31	27, 30	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷) → 𝑆 ≈ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )})
32	31	3adant3r 1200	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧:𝐴⟶1-1onto𝐶 ∧ (𝑦:𝐵⟶1-1onto𝐷 ∧ (𝑦‘ 0 ) = 𝑊)) → 𝑆 ≈ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )})
33		breq2 5112	. . . . . . . . . . 11 ⊢ ((𝑦‘ 0 ) = 𝑊 → (𝑥 finSupp (𝑦‘ 0 ) ↔ 𝑥 finSupp 𝑊))
34	33	rabbidv 3423	. . . . . . . . . 10 ⊢ ((𝑦‘ 0 ) = 𝑊 → {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊})
35		mapfien2.t	. . . . . . . . . 10 ⊢ 𝑇 = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp 𝑊}
36	34, 35	eqtr4di 2816	. . . . . . . . 9 ⊢ ((𝑦‘ 0 ) = 𝑊 → {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = 𝑇)
37	36	adantl 487	. . . . . . . 8 ⊢ ((𝑦:𝐵⟶1-1onto𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = 𝑇)
38	37	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧:𝐴⟶1-1onto𝐶 ∧ (𝑦:𝐵⟶1-1onto𝐷 ∧ (𝑦‘ 0 ) = 𝑊)) → {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = 𝑇)
39	32, 38	breqtrd 5137	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧:𝐴⟶1-1onto𝐶 ∧ (𝑦:𝐵⟶1-1onto𝐷 ∧ (𝑦‘ 0 ) = 𝑊)) → 𝑆 ≈ 𝑇)
40	39	3exp 1137	. . . . 5 ⊢ (𝜑 → (𝑧:𝐴⟶1-1onto𝐶 → ((𝑦:𝐵⟶1-1onto𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → 𝑆 ≈ 𝑇)))
41	40	exlimdv 1968	. . . 4 ⊢ (𝜑 → (∃𝑧 𝑧:𝐴⟶1-1onto𝐶 → ((𝑦:𝐵⟶1-1onto𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → 𝑆 ≈ 𝑇)))
42	8, 41	mpd 16	. . 3 ⊢ (𝜑 → ((𝑦:𝐵⟶1-1onto𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → 𝑆 ≈ 𝑇))
43	42	exlimdv 1968	. 2 ⊢ (𝜑 → (∃𝑦(𝑦:𝐵⟶1-1onto𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → 𝑆 ≈ 𝑇))
44	5, 43	mpd 16	1 ⊢ (𝜑 → 𝑆 ≈ 𝑇)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {crab 3416  Vcvv 3454   class class class wbr 5108   ↦ cmpt 5192  ^◡ccnv 5657   ∘ ccom 5662  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451   ↑_m cmap 8885   ≈ cen 9008   finSupp cfsupp 9391

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-supp 8215  df-1o 8514  df-map 8887  df-en 9012  df-dom 9013  df-fin 9015  df-fsupp 9392

This theorem is used by:  frlmpwfi  44366