Theorem mapfien2 9312

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 9014	. . 3 ⊢ (( 0 ∈ 𝐵 ∧ 𝑊 ∈ 𝐷 ∧ 𝐵 ≈ 𝐷) → ∃𝑦(𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊))
5	1, 2, 3, 4	syl3anc 1373	. 2 ⊢ (𝜑 → ∃𝑦(𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊))
6		mapfien2.ac	. . . . 5 ⊢ (𝜑 → 𝐴 ≈ 𝐶)
7		bren 8893	. . . . 5 ⊢ (𝐴 ≈ 𝐶 ↔ ∃𝑧 𝑧:𝐴–1-1-onto→𝐶)
8	6, 7	sylib 218	. . . 4 ⊢ (𝜑 → ∃𝑧 𝑧:𝐴–1-1-onto→𝐶)
9		mapfien2.s	. . . . . . . . . 10 ⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 0 }
10		eqid 2736	. . . . . . . . . 10 ⊢ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )}
11		eqid 2736	. . . . . . . . . 10 ⊢ (𝑦‘ 0 ) = (𝑦‘ 0 )
12		f1ocnv 6786	. . . . . . . . . . 11 ⊢ (𝑧:𝐴–1-1-onto→𝐶 → ◡𝑧:𝐶–1-1-onto→𝐴)
13	12	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → ◡𝑧:𝐶–1-1-onto→𝐴)
14		simp3 1138	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝑦:𝐵–1-1-onto→𝐷)
15	6	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐴 ≈ 𝐶)
16		relen 8888	. . . . . . . . . . . 12 ⊢ Rel ≈
17	16	brrelex1i 5680	. . . . . . . . . . 11 ⊢ (𝐴 ≈ 𝐶 → 𝐴 ∈ V)
18	15, 17	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐴 ∈ V)
19	3	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐵 ≈ 𝐷)
20	16	brrelex1i 5680	. . . . . . . . . . 11 ⊢ (𝐵 ≈ 𝐷 → 𝐵 ∈ V)
21	19, 20	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐵 ∈ V)
22	16	brrelex2i 5681	. . . . . . . . . . 11 ⊢ (𝐴 ≈ 𝐶 → 𝐶 ∈ V)
23	15, 22	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐶 ∈ V)
24	16	brrelex2i 5681	. . . . . . . . . . 11 ⊢ (𝐵 ≈ 𝐷 → 𝐷 ∈ V)
25	19, 24	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐷 ∈ V)
26	1	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 0 ∈ 𝐵)
27	9, 10, 11, 13, 14, 18, 21, 23, 25, 26	mapfien 9311	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → (𝑤 ∈ 𝑆 ↦ (𝑦 ∘ (𝑤 ∘ ◡𝑧))):𝑆–1-1-onto→{𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )})
28		ovex 7391	. . . . . . . . . . 11 ⊢ (𝐵 ↑m 𝐴) ∈ V
29	9, 28	rabex2 5286	. . . . . . . . . 10 ⊢ 𝑆 ∈ V
30	29	f1oen 8909	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝑆 ↦ (𝑦 ∘ (𝑤 ∘ ◡𝑧))):𝑆–1-1-onto→{𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} → 𝑆 ≈ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )})
31	27, 30	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝑆 ≈ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )})
32	31	3adant3r 1182	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ (𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊)) → 𝑆 ≈ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )})
33		breq2 5102	. . . . . . . . . . 11 ⊢ ((𝑦‘ 0 ) = 𝑊 → (𝑥 finSupp (𝑦‘ 0 ) ↔ 𝑥 finSupp 𝑊))
34	33	rabbidv 3406	. . . . . . . . . 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 2789	. . . . . . . . 9 ⊢ ((𝑦‘ 0 ) = 𝑊 → {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = 𝑇)
37	36	adantl 481	. . . . . . . 8 ⊢ ((𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = 𝑇)
38	37	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ (𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊)) → {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = 𝑇)
39	32, 38	breqtrd 5124	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ (𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊)) → 𝑆 ≈ 𝑇)
40	39	3exp 1119	. . . . 5 ⊢ (𝜑 → (𝑧:𝐴–1-1-onto→𝐶 → ((𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → 𝑆 ≈ 𝑇)))
41	40	exlimdv 1934	. . . 4 ⊢ (𝜑 → (∃𝑧 𝑧:𝐴–1-1-onto→𝐶 → ((𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → 𝑆 ≈ 𝑇)))
42	8, 41	mpd 15	. . 3 ⊢ (𝜑 → ((𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → 𝑆 ≈ 𝑇))
43	42	exlimdv 1934	. 2 ⊢ (𝜑 → (∃𝑦(𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → 𝑆 ≈ 𝑇))
44	5, 43	mpd 15	1 ⊢ (𝜑 → 𝑆 ≈ 𝑇)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {crab 3399  Vcvv 3440   class class class wbr 5098   ↦ cmpt 5179  ^◡ccnv 5623   ∘ ccom 5628  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763   ≈ cen 8880   finSupp cfsupp 9264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-1o 8397  df-map 8765  df-en 8884  df-dom 8885  df-fin 8887  df-fsupp 9265

This theorem is referenced by:  frlmpwfi  43340