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Theorem iunxprg 4537
Description: A pair index picks out two instances of an indexed union's argument. (Contributed by Alexander van der Vekens, 2-Feb-2018.)
Hypotheses
Ref Expression
iunxprg.1 (𝑥 = 𝐴𝐶 = 𝐷)
iunxprg.2 (𝑥 = 𝐵𝐶 = 𝐸)
Assertion
Ref Expression
iunxprg ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐸
Allowed substitution hints:   𝐶(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem iunxprg
StepHypRef Expression
1 df-pr 4127 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2 iuneq1 4464 . . . 4 ({𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶)
31, 2ax-mp 5 . . 3 𝑥 ∈ {𝐴, 𝐵}𝐶 = 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶
4 iunxun 4535 . . 3 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶 = ( 𝑥 ∈ {𝐴}𝐶 𝑥 ∈ {𝐵}𝐶)
53, 4eqtri 2631 . 2 𝑥 ∈ {𝐴, 𝐵}𝐶 = ( 𝑥 ∈ {𝐴}𝐶 𝑥 ∈ {𝐵}𝐶)
6 iunxprg.1 . . . . 5 (𝑥 = 𝐴𝐶 = 𝐷)
76iunxsng 4532 . . . 4 (𝐴𝑉 𝑥 ∈ {𝐴}𝐶 = 𝐷)
87adantr 479 . . 3 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴}𝐶 = 𝐷)
9 iunxprg.2 . . . . 5 (𝑥 = 𝐵𝐶 = 𝐸)
109iunxsng 4532 . . . 4 (𝐵𝑊 𝑥 ∈ {𝐵}𝐶 = 𝐸)
1110adantl 480 . . 3 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐵}𝐶 = 𝐸)
128, 11uneq12d 3729 . 2 ((𝐴𝑉𝐵𝑊) → ( 𝑥 ∈ {𝐴}𝐶 𝑥 ∈ {𝐵}𝐶) = (𝐷𝐸))
135, 12syl5eq 2655 1 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cun 3537  {csn 4124  {cpr 4126   ciun 4449
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-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
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-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174  df-sbc 3402  df-un 3544  df-in 3546  df-ss 3553  df-sn 4125  df-pr 4127  df-iun 4451
This theorem is referenced by:  ovnsubadd2lem  39332  rnfdmpr  40134
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