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Theorem fmptpr 6320
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 4127 . . 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 6317 . . 3 (𝜑 → {⟨𝐴, 𝐶⟩} = (𝑥 ∈ {𝐴} ↦ 𝐸))
76uneq1d 3727 . 2 (𝜑 → ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = ((𝑥 ∈ {𝐴} ↦ 𝐸) ∪ {⟨𝐵, 𝐷⟩}))
8 fmptpr.2 . . . 4 (𝜑𝐵𝑊)
9 elex 3184 . . . 4 (𝐵𝑊𝐵 ∈ V)
108, 9syl 17 . . 3 (𝜑𝐵 ∈ V)
11 fmptpr.4 . . . 4 (𝜑𝐷𝑌)
12 elex 3184 . . . 4 (𝐷𝑌𝐷 ∈ V)
1311, 12syl 17 . . 3 (𝜑𝐷 ∈ V)
14 df-pr 4127 . . . . 5 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
1514eqcomi 2618 . . . 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 6319 . 2 (𝜑 → ((𝑥 ∈ {𝐴} ↦ 𝐸) ∪ {⟨𝐵, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸))
192, 7, 183eqtrd 2647 1 (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐵} ↦ 𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  cun 3537  {csn 4124  {cpr 4126  cop 4130  cmpt 4637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-opab 4638  df-mpt 4639
This theorem is referenced by:  pmtrprfvalrn  17679  esumsnf  29246  sge0sn  39055  zlmodzxzscm  41909  zlmodzxzadd  41910
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