Theorem oef1o 9689

Description: A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 7297.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)

Hypotheses

Ref	Expression
oef1o.f	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶)
oef1o.g	⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
oef1o.a	⊢ (𝜑 → 𝐴 ∈ (On ∖ 1o))
oef1o.b	⊢ (𝜑 → 𝐵 ∈ On)
oef1o.c	⊢ (𝜑 → 𝐶 ∈ On)
oef1o.d	⊢ (𝜑 → 𝐷 ∈ On)
oef1o.z	⊢ (𝜑 → (𝐹‘∅) = ∅)
oef1o.k	⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺)))
oef1o.h	⊢ 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵))

Assertion

Ref	Expression
oef1o	⊢ (𝜑 → 𝐻:(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦

Allowed substitution hints:   𝐻(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem oef1o

Step	Hyp	Ref	Expression
1		eqid 2732	. . . . 5 ⊢ dom (𝐶 CNF 𝐷) = dom (𝐶 CNF 𝐷)
2		oef1o.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ On)
3		oef1o.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ On)
4	1, 2, 3	cantnff1o 9687	. . . 4 ⊢ (𝜑 → (𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)–1-1-onto→(𝐶 ↑o 𝐷))
5		eqid 2732	. . . . . . . 8 ⊢ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}
6		eqid 2732	. . . . . . . 8 ⊢ {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}
7		eqid 2732	. . . . . . . 8 ⊢ (𝐹‘∅) = (𝐹‘∅)
8		oef1o.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
9		f1ocnv 6842	. . . . . . . . 9 ⊢ (𝐺:𝐵–1-1-onto→𝐷 → ◡𝐺:𝐷–1-1-onto→𝐵)
10	8, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → ◡𝐺:𝐷–1-1-onto→𝐵)
11		oef1o.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶)
12		oef1o.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ On)
13		oef1o.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 1o))
14		ondif1 8497	. . . . . . . . . 10 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
15	14	simprbi 497	. . . . . . . . 9 ⊢ (𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴)
16	13, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐴)
17	5, 6, 7, 10, 11, 12, 13, 3, 2, 16	mapfien 9399	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺))):{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
18		oef1o.k	. . . . . . . 8 ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺)))
19		f1oeq1 6818	. . . . . . . 8 ⊢ (𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺))) → (𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} ↔ (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺))):{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}))
20	18, 19	ax-mp 5	. . . . . . 7 ⊢ (𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} ↔ (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺))):{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
21	17, 20	sylibr 233	. . . . . 6 ⊢ (𝜑 → 𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
22		eqid 2732	. . . . . . . . 9 ⊢ {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅}
23	22, 2, 3	cantnfdm 9655	. . . . . . . 8 ⊢ (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅})
24		oef1o.z	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘∅) = ∅)
25	24	breq2d 5159	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 finSupp (𝐹‘∅) ↔ 𝑥 finSupp ∅))
26	25	rabbidv 3440	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅})
27	23, 26	eqtr4d 2775	. . . . . . 7 ⊢ (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
28	27	f1oeq3d 6827	. . . . . 6 ⊢ (𝜑 → (𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷) ↔ 𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}))
29	21, 28	mpbird 256	. . . . 5 ⊢ (𝜑 → 𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷))
30	13	eldifad 3959	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
31	5, 30, 12	cantnfdm 9655	. . . . . 6 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅})
32	31	f1oeq2d 6826	. . . . 5 ⊢ (𝜑 → (𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷) ↔ 𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷)))
33	29, 32	mpbird 256	. . . 4 ⊢ (𝜑 → 𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷))
34		f1oco 6853	. . . 4 ⊢ (((𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)–1-1-onto→(𝐶 ↑o 𝐷) ∧ 𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷)) → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))
35	4, 33, 34	syl2anc 584	. . 3 ⊢ (𝜑 → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))
36		eqid 2732	. . . . 5 ⊢ dom (𝐴 CNF 𝐵) = dom (𝐴 CNF 𝐵)
37	36, 30, 12	cantnff1o 9687	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐴 ↑o 𝐵))
38		f1ocnv 6842	. . . 4 ⊢ ((𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐴 ↑o 𝐵) → ◡(𝐴 CNF 𝐵):(𝐴 ↑o 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵))
39	37, 38	syl 17	. . 3 ⊢ (𝜑 → ◡(𝐴 CNF 𝐵):(𝐴 ↑o 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵))
40		f1oco 6853	. . 3 ⊢ ((((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶 ↑o 𝐷) ∧ ◡(𝐴 CNF 𝐵):(𝐴 ↑o 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵)) → (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)):(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))
41	35, 39, 40	syl2anc 584	. 2 ⊢ (𝜑 → (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)):(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))
42		oef1o.h	. . 3 ⊢ 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵))
43		f1oeq1 6818	. . 3 ⊢ (𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)) → (𝐻:(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷) ↔ (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)):(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷)))
44	42, 43	ax-mp 5	. 2 ⊢ (𝐻:(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷) ↔ (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)):(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))
45	41, 44	sylibr 233	1 ⊢ (𝜑 → 𝐻:(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  {crab 3432   ∖ cdif 3944  ∅c0 4321   class class class wbr 5147   ↦ cmpt 5230  ^◡ccnv 5674  dom cdm 5675   ∘ ccom 5679  Oncon0 6361  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7405  1_oc1o 8455   ↑_o coe 8461   ↑_m cmap 8816   finSupp cfsupp 9357   CNF ccnf 9652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-seqom 8444  df-1o 8462  df-2o 8463  df-oadd 8466  df-omul 8467  df-oexp 8468  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-oi 9501  df-cnf 9653

This theorem is referenced by:  infxpenc  10009