Theorem oef1o 8595

Description: A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 6557.) (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 2622	. . . . 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 8593	. . . 4 ⊢ (𝜑 → (𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)–1-1-onto→(𝐶 ↑𝑜 𝐷))
5		eqid 2622	. . . . . . . 8 ⊢ {𝑥 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑥 finSupp ∅}
6		eqid 2622	. . . . . . . 8 ⊢ {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}
7		eqid 2622	. . . . . . . 8 ⊢ (𝐹‘∅) = (𝐹‘∅)
8		oef1o.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
9		f1ocnv 6149	. . . . . . . . 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		ssv 3625	. . . . . . . . 9 ⊢ On ⊆ V
13		oef1o.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ On)
14	12, 13	sseldi 3601	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ V)
15		oef1o.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 1𝑜))
16	15	eldifad 3586	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ On)
17	12, 16	sseldi 3601	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ V)
18	12, 3	sseldi 3601	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ V)
19	12, 2	sseldi 3601	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ V)
20		ondif1 7581	. . . . . . . . . 10 ⊢ (𝐴 ∈ (On ∖ 1𝑜) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
21	20	simprbi 480	. . . . . . . . 9 ⊢ (𝐴 ∈ (On ∖ 1𝑜) → ∅ ∈ 𝐴)
22	15, 21	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐴)
23	5, 6, 7, 10, 11, 14, 17, 18, 19, 22	mapfien 8313	. . . . . . 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‘∅)})
24		oef1o.k	. . . . . . . 8 ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺)))
25		f1oeq1 6127	. . . . . . . 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‘∅)}))
26	24, 25	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‘∅)})
27	23, 26	sylibr 224	. . . . . 6 ⊢ (𝜑 → 𝐾:{𝑥 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
28		eqid 2622	. . . . . . . . 9 ⊢ {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp ∅}
29	28, 2, 3	cantnfdm 8561	. . . . . . . 8 ⊢ (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp ∅})
30		oef1o.z	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘∅) = ∅)
31	30	breq2d 4665	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 finSupp (𝐹‘∅) ↔ 𝑥 finSupp ∅))
32	31	rabbidv 3189	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp ∅})
33	29, 32	eqtr4d 2659	. . . . . . 7 ⊢ (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
34		f1oeq3 6129	. . . . . . 7 ⊢ (dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶 ↑𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} → (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‘∅)}))
35	33, 34	syl 17	. . . . . 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‘∅)}))
36	27, 35	mpbird 247	. . . . 5 ⊢ (𝜑 → 𝐾:{𝑥 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷))
37	5, 16, 13	cantnfdm 8561	. . . . . 6 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑥 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑥 finSupp ∅})
38		f1oeq2 6128	. . . . . 6 ⊢ (dom (𝐴 CNF 𝐵) = {𝑥 ∈ (𝐴 ↑𝑚 𝐵) ∣ 𝑥 finSupp ∅} → (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)))
39	37, 38	syl 17	. . . . 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)))
40	36, 39	mpbird 247	. . . 4 ⊢ (𝜑 → 𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷))
41		f1oco 6159	. . . 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))
42	4, 40, 41	syl2anc 693	. . 3 ⊢ (𝜑 → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶 ↑𝑜 𝐷))
43		eqid 2622	. . . . 5 ⊢ dom (𝐴 CNF 𝐵) = dom (𝐴 CNF 𝐵)
44	43, 16, 13	cantnff1o 8593	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐴 ↑𝑜 𝐵))
45		f1ocnv 6149	. . . 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))
46	44, 45	syl 17	. . 3 ⊢ (𝜑 → ◡(𝐴 CNF 𝐵):(𝐴 ↑𝑜 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵))
47		f1oco 6159	. . 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))
48	42, 46, 47	syl2anc 693	. 2 ⊢ (𝜑 → (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵)):(𝐴 ↑𝑜 𝐵)–1-1-onto→(𝐶 ↑𝑜 𝐷))
49		oef1o.h	. . 3 ⊢ 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵))
50		f1oeq1 6127	. . 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)))
51	49, 50	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))
52	48, 51	sylibr 224	1 ⊢ (𝜑 → 𝐻:(𝐴 ↑𝑜 𝐵)–1-1-onto→(𝐶 ↑𝑜 𝐷))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   ∖ cdif 3571  ∅c0 3915   class class class wbr 4653   ↦ cmpt 4729  ^◡ccnv 5113  dom cdm 5114   ∘ ccom 5118  Oncon0 5723  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553   ↑_𝑜 coe 7559   ↑_𝑚 cmap 7857   finSupp cfsupp 8275   CNF ccnf 8558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-cnf 8559

This theorem is referenced by:  infxpenc  8841