MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oef1o Structured version   Visualization version   GIF version

Theorem oef1o 9641
Description: A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 7254.) (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 2737 . . . . 5 dom (𝐶 CNF 𝐷) = dom (𝐶 CNF 𝐷)
2 oef1o.c . . . . 5 (𝜑𝐶 ∈ On)
3 oef1o.d . . . . 5 (𝜑𝐷 ∈ On)
41, 2, 3cantnff1o 9639 . . . 4 (𝜑 → (𝐶 CNF 𝐷):dom (𝐶 CNF 𝐷)–1-1-onto→(𝐶o 𝐷))
5 eqid 2737 . . . . . . . 8 {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}
6 eqid 2737 . . . . . . . 8 {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)}
7 eqid 2737 . . . . . . . 8 (𝐹‘∅) = (𝐹‘∅)
8 oef1o.g . . . . . . . . 9 (𝜑𝐺:𝐵1-1-onto𝐷)
9 f1ocnv 6801 . . . . . . . . 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 oef1o.b . . . . . . . 8 (𝜑𝐵 ∈ On)
13 oef1o.a . . . . . . . 8 (𝜑𝐴 ∈ (On ∖ 1o))
14 ondif1 8452 . . . . . . . . . 10 (𝐴 ∈ (On ∖ 1o) ↔ (𝐴 ∈ On ∧ ∅ ∈ 𝐴))
1514simprbi 498 . . . . . . . . 9 (𝐴 ∈ (On ∖ 1o) → ∅ ∈ 𝐴)
1613, 15syl 17 . . . . . . . 8 (𝜑 → ∅ ∈ 𝐴)
175, 6, 7, 10, 11, 12, 13, 3, 2, 16mapfien 9351 . . . . . . 7 (𝜑 → (𝑦 ∈ {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺))):{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
18 oef1o.k . . . . . . . 8 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦𝐺)))
19 f1oeq1 6777 . . . . . . . 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‘∅)}))
2018, 19ax-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‘∅)})
2117, 20sylibr 233 . . . . . 6 (𝜑𝐾:{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→{𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
22 eqid 2737 . . . . . . . . 9 {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp ∅}
2322, 2, 3cantnfdm 9607 . . . . . . . 8 (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp ∅})
24 oef1o.z . . . . . . . . . 10 (𝜑 → (𝐹‘∅) = ∅)
2524breq2d 5122 . . . . . . . . 9 (𝜑 → (𝑥 finSupp (𝐹‘∅) ↔ 𝑥 finSupp ∅))
2625rabbidv 3418 . . . . . . . 8 (𝜑 → {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)} = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp ∅})
2723, 26eqtr4d 2780 . . . . . . 7 (𝜑 → dom (𝐶 CNF 𝐷) = {𝑥 ∈ (𝐶m 𝐷) ∣ 𝑥 finSupp (𝐹‘∅)})
2827f1oeq3d 6786 . . . . . 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‘∅)}))
2921, 28mpbird 257 . . . . 5 (𝜑𝐾:{𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅}–1-1-onto→dom (𝐶 CNF 𝐷))
3013eldifad 3927 . . . . . . 7 (𝜑𝐴 ∈ On)
315, 30, 12cantnfdm 9607 . . . . . 6 (𝜑 → dom (𝐴 CNF 𝐵) = {𝑥 ∈ (𝐴m 𝐵) ∣ 𝑥 finSupp ∅})
3231f1oeq2d 6785 . . . . 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)))
3329, 32mpbird 257 . . . 4 (𝜑𝐾:dom (𝐴 CNF 𝐵)–1-1-onto→dom (𝐶 CNF 𝐷))
34 f1oco 6812 . . . 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))
354, 33, 34syl2anc 585 . . 3 (𝜑 → ((𝐶 CNF 𝐷) ∘ 𝐾):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐶o 𝐷))
36 eqid 2737 . . . . 5 dom (𝐴 CNF 𝐵) = dom (𝐴 CNF 𝐵)
3736, 30, 12cantnff1o 9639 . . . 4 (𝜑 → (𝐴 CNF 𝐵):dom (𝐴 CNF 𝐵)–1-1-onto→(𝐴o 𝐵))
38 f1ocnv 6801 . . . 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))
3937, 38syl 17 . . 3 (𝜑(𝐴 CNF 𝐵):(𝐴o 𝐵)–1-1-onto→dom (𝐴 CNF 𝐵))
40 f1oco 6812 . . 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))
4135, 39, 40syl2anc 585 . 2 (𝜑 → (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵)):(𝐴o 𝐵)–1-1-onto→(𝐶o 𝐷))
42 oef1o.h . . 3 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ (𝐴 CNF 𝐵))
43 f1oeq1 6777 . . 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)))
4442, 43ax-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))
4541, 44sylibr 233 1 (𝜑𝐻:(𝐴o 𝐵)–1-1-onto→(𝐶o 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  {crab 3410  cdif 3912  c0 4287   class class class wbr 5110  cmpt 5193  ccnv 5637  dom cdm 5638  ccom 5642  Oncon0 6322  1-1-ontowf1o 6500  cfv 6501  (class class class)co 7362  1oc1o 8410  o coe 8416  m cmap 8772   finSupp cfsupp 9312   CNF ccnf 9604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-supp 8098  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-seqom 8399  df-1o 8417  df-2o 8418  df-oadd 8421  df-omul 8422  df-oexp 8423  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-fsupp 9313  df-oi 9453  df-cnf 9605
This theorem is referenced by:  infxpenc  9961
  Copyright terms: Public domain W3C validator