Theorem fvun2d 5766

Description: The value of a union when the argument is in the second domain, a deduction version. (Contributed by metakunt, 28-May-2024.)

Hypotheses

Ref	Expression
fvun2d.1	|- ( ph -> F Fn A )
fvun2d.2	|- ( ph -> G Fn B )
fvun2d.3	|- ( ph -> ( A i^i B ) = (/) )
fvun2d.4	|- ( ph -> X e. B )

Assertion

Ref	Expression
fvun2d	|- ( ph -> ( ( F u. G ) ` X ) = ( G ` X ) )

Proof of Theorem fvun2d

Step	Hyp	Ref	Expression
1		fvun2d.1	. . 3 |- ( ph -> F Fn A )
2		fvun2d.2	. . 3 |- ( ph -> G Fn B )
3		fvun2d.3	. . . 4 |- ( ph -> ( A i^i B ) = (/) )
4		fvun2d.4	. . . 4 |- ( ph -> X e. B )
5	3, 4	jca 306	. . 3 |- ( ph -> ( ( A i^i B ) = (/) /\ X e. B ) )
6	1, 2, 5	3jca 1208	. 2 |- ( ph -> ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. B ) ) )
7		fvun2 5764	. 2 |- ( ( F Fn A /\ G Fn B /\ ( ( A i^i B ) = (/) /\ X e. B ) ) -> ( ( F u. G ) ` X ) = ( G ` X ) )
8	6, 7	syl 14	1 |- ( ph -> ( ( F u. G ) ` X ) = ( G ` X ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   u. cun 3218   i^i cin 3219   (/)c0 3520   Fn wfn 5367   ` cfv 5372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380

This theorem is referenced by: (None)