Theorem mapfien2 9407

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 9084	. . 3 ⊢ (( 0 ∈ 𝐵 ∧ 𝑊 ∈ 𝐷 ∧ 𝐵 ≈ 𝐷) → ∃𝑦(𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊))
5	1, 2, 3, 4	syl3anc 1370	. 2 ⊢ (𝜑 → ∃𝑦(𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊))
6		mapfien2.ac	. . . . 5 ⊢ (𝜑 → 𝐴 ≈ 𝐶)
7		bren 8952	. . . . 5 ⊢ (𝐴 ≈ 𝐶 ↔ ∃𝑧 𝑧:𝐴–1-1-onto→𝐶)
8	6, 7	sylib 217	. . . 4 ⊢ (𝜑 → ∃𝑧 𝑧:𝐴–1-1-onto→𝐶)
9		mapfien2.s	. . . . . . . . . 10 ⊢ 𝑆 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 0 }
10		eqid 2731	. . . . . . . . . 10 ⊢ {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )} = {𝑥 ∈ (𝐷 ↑m 𝐶) ∣ 𝑥 finSupp (𝑦‘ 0 )}
11		eqid 2731	. . . . . . . . . 10 ⊢ (𝑦‘ 0 ) = (𝑦‘ 0 )
12		f1ocnv 6846	. . . . . . . . . . 11 ⊢ (𝑧:𝐴–1-1-onto→𝐶 → ◡𝑧:𝐶–1-1-onto→𝐴)
13	12	3ad2ant2 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → ◡𝑧:𝐶–1-1-onto→𝐴)
14		simp3 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝑦:𝐵–1-1-onto→𝐷)
15	6	3ad2ant1 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐴 ≈ 𝐶)
16		relen 8947	. . . . . . . . . . . 12 ⊢ Rel ≈
17	16	brrelex1i 5733	. . . . . . . . . . 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 1132	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → 𝐵 ≈ 𝐷)
20	16	brrelex1i 5733	. . . . . . . . . . 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 5734	. . . . . . . . . . 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 5734	. . . . . . . . . . 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 1132	. . . . . . . . . 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 9406	. . . . . . . . 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 7445	. . . . . . . . . . 11 ⊢ (𝐵 ↑m 𝐴) ∈ V
29	9, 28	rabex2 5335	. . . . . . . . . 10 ⊢ 𝑆 ∈ V
30	29	f1oen 8972	. . . . . . . . 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 1180	. . . . . . 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 5153	. . . . . . . . . . 11 ⊢ ((𝑦‘ 0 ) = 𝑊 → (𝑥 finSupp (𝑦‘ 0 ) ↔ 𝑥 finSupp 𝑊))
34	33	rabbidv 3439	. . . . . . . . . 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 1134	. . . . . . 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 5175	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧:𝐴–1-1-onto→𝐶 ∧ (𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊)) → 𝑆 ≈ 𝑇)
40	39	3exp 1118	. . . . 5 ⊢ (𝜑 → (𝑧:𝐴–1-1-onto→𝐶 → ((𝑦:𝐵–1-1-onto→𝐷 ∧ (𝑦‘ 0 ) = 𝑊) → 𝑆 ≈ 𝑇)))
41	40	exlimdv 1935	. . . 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 1935	. 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 1540  ∃wex 1780   ∈ wcel 2105  {crab 3431  Vcvv 3473   class class class wbr 5149   ↦ cmpt 5232  ^◡ccnv 5676   ∘ ccom 5681  –1-1-onto→wf1o 6543  ‘cfv 6544  (class class class)co 7412   ↑_m cmap 8823   ≈ cen 8939   finSupp cfsupp 9364

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-supp 8150  df-1o 8469  df-map 8825  df-en 8943  df-fin 8946  df-fsupp 9365

This theorem is referenced by:  frlmpwfi  42143