Theorem funALTVss 39283

Description: Subclass theorem for function. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) (Revised by Peter Mazsa, 22-Sep-2021.)

Assertion

Ref	Expression
funALTVss	⊢ (𝐴 ⊆ 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))

Proof of Theorem funALTVss

Step	Hyp	Ref	Expression
1		cossss 39014	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵)
2		sstr2 3943	. . . 4 ⊢ ( ≀ 𝐴 ⊆ ≀ 𝐵 → ( ≀ 𝐵 ⊆ I → ≀ 𝐴 ⊆ I ))
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ( ≀ 𝐵 ⊆ I → ≀ 𝐴 ⊆ I ))
4		relss 5754	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴))
5	3, 4	anim12d 618	. 2 ⊢ (𝐴 ⊆ 𝐵 → (( ≀ 𝐵 ⊆ I ∧ Rel 𝐵) → ( ≀ 𝐴 ⊆ I ∧ Rel 𝐴)))
6		dffunALTV2 39272	. 2 ⊢ ( FunALTV 𝐵 ↔ ( ≀ 𝐵 ⊆ I ∧ Rel 𝐵))
7		dffunALTV2 39272	. 2 ⊢ ( FunALTV 𝐴 ↔ ( ≀ 𝐴 ⊆ I ∧ Rel 𝐴))
8	5, 6, 7	3imtr4g 298	1 ⊢ (𝐴 ⊆ 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))

Syntax hints:   → wi 4   ∧ wa 399   ⊆ wss 3904   I cid 5541  Rel wrel 5652   ≀ ccoss 38682   FunALTV wfunALTV 38715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-coss 39000  df-cnvrefrel 39106  df-funALTV 39266

This theorem is referenced by:  funALTVeq  39284  disjss  39330