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Theorem fmptpr 6479
Description: Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)
Hypotheses
Ref Expression
fmptpr.1 (𝜑𝐴𝑉)
fmptpr.2 (𝜑𝐵𝑊)
fmptpr.3 (𝜑𝐶𝑋)
fmptpr.4 (𝜑𝐷𝑌)
fmptpr.5 ((𝜑𝑥 = 𝐴) → 𝐸 = 𝐶)
fmptpr.6 ((𝜑𝑥 = 𝐵) → 𝐸 = 𝐷)
Assertion
Ref Expression
fmptpr (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝜑,𝑥
Allowed substitution hints:   𝐸(𝑥)   𝑉(𝑥)   𝑊(𝑥)   𝑋(𝑥)   𝑌(𝑥)

Proof of Theorem fmptpr
StepHypRef Expression
1 df-pr 4213 . . 3 {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
21a1i 11 . 2 (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}))
3 fmptpr.5 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐸 = 𝐶)
4 fmptpr.1 . . . 4 (𝜑𝐴𝑉)
5 fmptpr.3 . . . 4 (𝜑𝐶𝑋)
63, 4, 5fmptsnd 6476 . . 3 (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐸))
76uneq1d 3799 . 2 (𝜑 → ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = ((𝑥 ∈ {𝐴} ↦ 𝐸) ∪ {⟨𝐵, 𝐷⟩}))
8 fmptpr.2 . . . 4 (𝜑𝐵𝑊)
9 elex 3243 . . . 4 (𝐵𝑊𝐵 ∈ V)
108, 9syl 17 . . 3 (𝜑𝐵 ∈ V)
11 fmptpr.4 . . . 4 (𝜑𝐷𝑌)
12 elex 3243 . . . 4 (𝐷𝑌𝐷 ∈ V)
1311, 12syl 17 . . 3 (𝜑𝐷 ∈ V)
14 df-pr 4213 . . . . 5 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
1514eqcomi 2660 . . . 4 ({𝐴} ∪ {𝐵}) = {𝐴, 𝐵}
1615a1i 11 . . 3 (𝜑 → ({𝐴} ∪ {𝐵}) = {𝐴, 𝐵})
17 fmptpr.6 . . 3 ((𝜑𝑥 = 𝐵) → 𝐸 = 𝐷)
1810, 13, 16, 17fmptapd 6478 . 2 (𝜑 → ((𝑥 ∈ {𝐴} ↦ 𝐸) ∪ {⟨𝐵, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸))
192, 7, 183eqtrd 2689 1 (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  {csn 4210  {cpr 4212  cop 4216  cmpt 4762
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-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  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-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-mpt 4763
This theorem is referenced by:  pmtrprfvalrn  17954  esumsnf  30254  sge0sn  40914  zlmodzxzscm  42460  zlmodzxzadd  42461
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