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Theorem oef1o 8633
Description: A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 6597.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
Hypotheses
Ref Expression
oef1o.f (𝜑𝐹:𝐴1-1-onto𝐶)
oef1o.g (𝜑𝐺:𝐵1-1-onto𝐷)
oef1o.a (𝜑𝐴 ∈ (On ∖ 1𝑜))
oef1o.b (𝜑𝐵 ∈ On)
oef1o.c (𝜑𝐶 ∈ On)
oef1o.d (𝜑𝐷 ∈ On)
oef1o.z (𝜑 → (𝐹‘∅) = ∅)
oef1o.k 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺)))
oef1o.h 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵))
Assertion
Ref Expression
oef1o (𝜑𝐻:(𝐴𝑜 𝐵)–1-1-onto→(𝐶𝑜 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
StepHypRef Expression
1 eqid 2651 . . . . 5 dom (𝐶 CNF 𝐷) = dom (𝐶 CNF 𝐷)
2 oef1o.c . . . . 5 (𝜑𝐶 ∈ On)
3 oef1o.d . . . . 5 (𝜑𝐷 ∈ On)
41, 2, 3cantnff1o 8631 . . . 4 (𝜑 → (𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)–1-1-onto→(𝐶𝑜 𝐷))
5 eqid 2651 . . . . . . . 8 {𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅}
6 eqid 2651 . . . . . . . 8 {𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}
7 eqid 2651 . . . . . . . 8 (𝐹‘∅) = (𝐹‘∅)
8 oef1o.g . . . . . . . . 9 (𝜑𝐺:𝐵1-1-onto𝐷)
9 f1ocnv 6187 . . . . . . . . 9 (𝐺:𝐵1-1-onto𝐷𝐺:𝐷1-1-onto𝐵)
108, 9syl 17 . . . . . . . 8 (𝜑𝐺:𝐷1-1-onto𝐵)
11 oef1o.f . . . . . . . 8 (𝜑𝐹:𝐴1-1-onto𝐶)
12 ssv 3658 . . . . . . . . 9 On ⊆ V
13 oef1o.b . . . . . . . . 9 (𝜑𝐵 ∈ On)
1412, 13sseldi 3634 . . . . . . . 8 (𝜑𝐵 ∈ V)
15 oef1o.a . . . . . . . . . 10 (𝜑𝐴 ∈ (On ∖ 1𝑜))
1615eldifad 3619 . . . . . . . . 9 (𝜑𝐴 ∈ On)
1712, 16sseldi 3634 . . . . . . . 8 (𝜑𝐴 ∈ V)
1812, 3sseldi 3634 . . . . . . . 8 (𝜑𝐷 ∈ V)
1912, 2sseldi 3634 . . . . . . . 8 (𝜑𝐶 ∈ V)
20 ondif1 7626 . . . . . . . . . 10 (𝐴 ∈ (On ∖ 1𝑜) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
2120simprbi 479 . . . . . . . . 9 (𝐴 ∈ (On ∖ 1𝑜) → ∅ ∈ 𝐴)
2215, 21syl 17 . . . . . . . 8 (𝜑 → ∅ ∈ 𝐴)
235, 6, 7, 10, 11, 14, 17, 18, 19, 22mapfien 8354 . . . . . . 7 (𝜑 → (𝑦 ∈ {𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺))):{𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
24 oef1o.k . . . . . . . 8 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺)))
25 f1oeq1 6165 . . . . . . . 8 (𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺))) → (𝐾:{𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} ↔ (𝑦 ∈ {𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺))):{𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}))
2624, 25ax-mp 5 . . . . . . 7 (𝐾:{𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} ↔ (𝑦 ∈ {𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺))):{𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
2723, 26sylibr 224 . . . . . 6 (𝜑𝐾:{𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
28 eqid 2651 . . . . . . . . 9 {𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp ∅}
2928, 2, 3cantnfdm 8599 . . . . . . . 8 (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp ∅})
30 oef1o.z . . . . . . . . . 10 (𝜑 → (𝐹‘∅) = ∅)
3130breq2d 4697 . . . . . . . . 9 (𝜑 → (𝑥 finSupp (𝐹‘∅) ↔ 𝑥 finSupp ∅))
3231rabbidv 3220 . . . . . . . 8 (𝜑 → {𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp ∅})
3329, 32eqtr4d 2688 . . . . . . 7 (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
34 f1oeq3 6167 . . . . . . 7 (dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} → (𝐾:{𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷) ↔ 𝐾:{𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}))
3533, 34syl 17 . . . . . 6 (𝜑 → (𝐾:{𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷) ↔ 𝐾:{𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶𝑚 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}))
3627, 35mpbird 247 . . . . 5 (𝜑𝐾:{𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷))
375, 16, 13cantnfdm 8599 . . . . . 6 (𝜑 → dom (𝐴 CNF 𝐵) = {𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅})
38 f1oeq2 6166 . . . . . 6 (dom (𝐴 CNF 𝐵) = {𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅} → (𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷) ↔ 𝐾:{𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷)))
3937, 38syl 17 . . . . 5 (𝜑 → (𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷) ↔ 𝐾:{𝑥 ∈ (𝐴𝑚 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷)))
4036, 39mpbird 247 . . . 4 (𝜑𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷))
41 f1oco 6197 . . . 4 (((𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)–1-1-onto→(𝐶𝑜 𝐷) ∧ 𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷)) → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶𝑜 𝐷))
424, 40, 41syl2anc 694 . . 3 (𝜑 → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶𝑜 𝐷))
43 eqid 2651 . . . . 5 dom (𝐴 CNF 𝐵) = dom (𝐴 CNF 𝐵)
4443, 16, 13cantnff1o 8631 . . . 4 (𝜑 → (𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐴𝑜 𝐵))
45 f1ocnv 6187 . . . 4 ((𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐴𝑜 𝐵) → (𝐴 CNF 𝐵):(𝐴𝑜 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵))
4644, 45syl 17 . . 3 (𝜑(𝐴 CNF 𝐵):(𝐴𝑜 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵))
47 f1oco 6197 . . 3 ((((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶𝑜 𝐷) ∧ (𝐴 CNF 𝐵):(𝐴𝑜 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵)) → (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵)):(𝐴𝑜 𝐵)–1-1-onto→(𝐶𝑜 𝐷))
4842, 46, 47syl2anc 694 . 2 (𝜑 → (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵)):(𝐴𝑜 𝐵)–1-1-onto→(𝐶𝑜 𝐷))
49 oef1o.h . . 3 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵))
50 f1oeq1 6165 . . 3 (𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵)) → (𝐻:(𝐴𝑜 𝐵)–1-1-onto→(𝐶𝑜 𝐷) ↔ (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵)):(𝐴𝑜 𝐵)–1-1-onto→(𝐶𝑜 𝐷)))
5149, 50ax-mp 5 . 2 (𝐻:(𝐴𝑜 𝐵)–1-1-onto→(𝐶𝑜 𝐷) ↔ (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵)):(𝐴𝑜 𝐵)–1-1-onto→(𝐶𝑜 𝐷))
5248, 51sylibr 224 1 (𝜑𝐻:(𝐴𝑜 𝐵)–1-1-onto→(𝐶𝑜 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  {crab 2945  Vcvv 3231  cdif 3604  c0 3948   class class class wbr 4685  cmpt 4762  ccnv 5142  dom cdm 5143  ccom 5147  Oncon0 5761  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  1𝑜c1o 7598  𝑜 coe 7604  𝑚 cmap 7899   finSupp cfsupp 8316   CNF ccnf 8596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seqom 7588  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-oexp 7611  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-cnf 8597
This theorem is referenced by:  infxpenc  8879
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