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Theorem iunxprg 4766
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 4339 . . . 4 {𝐴, 𝐵} = ({𝐴} ∪ {𝐵})
2 iuneq1 4692 . . . 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 4764 . . 3 𝑥 ∈ ({𝐴} ∪ {𝐵})𝐶 = ( 𝑥 ∈ {𝐴}𝐶 𝑥 ∈ {𝐵}𝐶)
53, 4eqtri 2787 . 2 𝑥 ∈ {𝐴, 𝐵}𝐶 = ( 𝑥 ∈ {𝐴}𝐶 𝑥 ∈ {𝐵}𝐶)
6 iunxprg.1 . . . . 5 (𝑥 = 𝐴𝐶 = 𝐷)
76iunxsng 4760 . . . 4 (𝐴𝑉 𝑥 ∈ {𝐴}𝐶 = 𝐷)
87adantr 472 . . 3 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴}𝐶 = 𝐷)
9 iunxprg.2 . . . . 5 (𝑥 = 𝐵𝐶 = 𝐸)
109iunxsng 4760 . . . 4 (𝐵𝑊 𝑥 ∈ {𝐵}𝐶 = 𝐸)
1110adantl 473 . . 3 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐵}𝐶 = 𝐸)
128, 11uneq12d 3932 . 2 ((𝐴𝑉𝐵𝑊) → ( 𝑥 ∈ {𝐴}𝐶 𝑥 ∈ {𝐵}𝐶) = (𝐷𝐸))
135, 12syl5eq 2811 1 ((𝐴𝑉𝐵𝑊) → 𝑥 ∈ {𝐴, 𝐵}𝐶 = (𝐷𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  cun 3732  {csn 4336  {cpr 4338   ciun 4678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-sbc 3599  df-un 3739  df-in 3741  df-ss 3748  df-sn 4337  df-pr 4339  df-iun 4680
This theorem is referenced by:  ovnsubadd2lem  41523  rnfdmpr  42054
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