Theorem mptssALT 29474

Description: Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5454. (Contributed by Thierry Arnoux, 30-May-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
mptssALT	|- ( A C_ B -> ( x e. A |-> C ) C_ ( x e. B |-> C ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem mptssALT

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3597	. . . 4 |- ( A C_ B -> ( x e. A -> x e. B ) )
2	1	anim1d 588	. . 3 |- ( A C_ B -> ( ( x e. A /\ y = C ) -> ( x e. B /\ y = C ) ) )
3	2	ssopab2dv 5004	. 2 |- ( A C_ B -> { <. x , y >. | ( x e. A /\ y = C ) } C_ { <. x , y >. | ( x e. B /\ y = C ) } )
4		df-mpt 4730	. 2 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
5		df-mpt 4730	. 2 |- ( x e. B |-> C ) = { <. x , y >. | ( x e. B /\ y = C ) }
6	3, 4, 5	3sstr4g 3646	1 |- ( A C_ B -> ( x e. A |-> C ) C_ ( x e. B |-> C ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   {copab 4712   |-> cmpt 4729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-in 3581  df-ss 3588  df-opab 4713  df-mpt 4730

This theorem is referenced by: (None)