Theorem oef1o 9653

Description: A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 7286.) (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 2762	. . . . 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 9651	. . . 4 ⊢ (𝜑 → (𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)–1-1-onto→(𝐶 ↑o 𝐷))
5		eqid 2762	. . . . . . . 8 ⊢ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}
6		eqid 2762	. . . . . . . 8 ⊢ {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}
7		eqid 2762	. . . . . . . 8 ⊢ (𝐹‘∅) = (𝐹‘∅)
8		oef1o.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)
9		f1ocnv 6819	. . . . . . . . 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 8470	. . . . . . . . . 10 ⊢ (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
15	14	simprbi 501	. . . . . . . . 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 9354	. . . . . . 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 6794	. . . . . . . 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 236	. . . . . 6 ⊢ (𝜑 → 𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
22		eqid 2762	. . . . . . . . 9 ⊢ {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅}
23	22, 2, 3	cantnfdm 9619	. . . . . . . 8 ⊢ (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅})
24		oef1o.z	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘∅) = ∅)
25	24	breq2d 5112	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 finSupp (𝐹‘∅) ↔ 𝑥 finSupp ∅))
26	25	rabbidv 3421	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp ∅})
27	23, 26	eqtr4d 2800	. . . . . . 7 ⊢ (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶 ↑m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
28	27	f1oeq3d 6803	. . . . . 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 259	. . . . 5 ⊢ (𝜑 → 𝐾:{𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷))
30	13	eldifad 3916	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
31	5, 30, 12	cantnfdm 9619	. . . . . 6 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅})
32	31	f1oeq2d 6802	. . . . 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 259	. . . 4 ⊢ (𝜑 → 𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷))
34		f1oco 6830	. . . 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 593	. . 3 ⊢ (𝜑 → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))
36		eqid 2762	. . . . 5 ⊢ dom (𝐴 CNF 𝐵) = dom (𝐴 CNF 𝐵)
37	36, 30, 12	cantnff1o 9651	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐴 ↑o 𝐵))
38		f1ocnv 6819	. . . 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 6830	. . 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 593	. 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 6794	. . 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 236	1 ⊢ (𝜑 → 𝐻:(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1560   ∈ wcel 2142  {crab 3414   ∖ cdif 3901  ∅c0 4285   class class class wbr 5100   ↦ cmpt 5181  ^◡ccnv 5646  dom cdm 5647   ∘ ccom 5651  Oncon0 6346  –1-1-onto→wf1o 6520  ‘cfv 6521  (class class class)co 7396  1_oc1o 8430   ↑_o coe 8436   ↑_m cmap 8808   finSupp cfsupp 9307   CNF ccnf 9616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-seqom 8419  df-1o 8437  df-2o 8438  df-oadd 8441  df-omul 8442  df-oexp 8443  df-er 8678  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9308  df-oi 9458  df-cnf 9617

This theorem is referenced by:  infxpenc  9974