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Theorem mpt2mptx 5595
 Description: Express a two-argument function as a one-argument function, or vice-versa. In this version 𝐵(𝑥) is not assumed to be constant w.r.t 𝑥. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
mpt2mpt.1 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
Assertion
Ref Expression
mpt2mptx (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑦,𝐵,𝑧   𝑥,𝐶,𝑦   𝑧,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑧)   𝐷(𝑥,𝑦)

Proof of Theorem mpt2mptx
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-mpt 3820 . 2 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)}
2 df-mpt2 5517 . . 3 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)}
3 eliunxp 4475 . . . . . . 7 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
43anbi1i 431 . . . . . 6 ((𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶) ↔ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶))
5 19.41vv 1783 . . . . . 6 (∃𝑥𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶))
6 anass 381 . . . . . . . 8 (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)))
7 mpt2mpt.1 . . . . . . . . . . 11 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)
87eqeq2d 2051 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑤 = 𝐶𝑤 = 𝐷))
98anbi2d 437 . . . . . . . . 9 (𝑧 = ⟨𝑥, 𝑦⟩ → (((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
109pm5.32i 427 . . . . . . . 8 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
116, 10bitri 173 . . . . . . 7 (((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
12112exbii 1497 . . . . . 6 (∃𝑥𝑦((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)) ∧ 𝑤 = 𝐶) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
134, 5, 123bitr2i 197 . . . . 5 ((𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)))
1413opabbii 3824 . . . 4 {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷))}
15 dfoprab2 5552 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)} = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷))}
1614, 15eqtr4i 2063 . . 3 {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐷)}
172, 16eqtr4i 2063 . 2 (𝑥𝐴, 𝑦𝐵𝐷) = {⟨𝑧, 𝑤⟩ ∣ (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ∧ 𝑤 = 𝐶)}
181, 17eqtr4i 2063 1 (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {csn 3375  ⟨cop 3378  ∪ ciun 3657  {copab 3817   ↦ cmpt 3818   × cxp 4343  {coprab 5513   ↦ cmpt2 5514 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-iun 3659  df-opab 3819  df-mpt 3820  df-xp 4351  df-rel 4352  df-oprab 5516  df-mpt2 5517 This theorem is referenced by:  mpt2mpt  5596  mpt2mptsx  5823  dmmpt2ssx  5825  fmpt2x  5826
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