Theorem funimass3 5825

Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 5824 would be the special case of 𝐴 being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)

Assertion

Ref	Expression
funimass3	⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵)))

Proof of Theorem funimass3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funimass4 5753	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
2		ssel 3242	. . . . . 6 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑥 ∈ 𝐴 → 𝑥 ∈ dom 𝐹))
3		fvimacnv 5824	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑥 ∈ (◡𝐹 “ 𝐵)))
4	3	ex 115	. . . . . 6 ⊢ (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑥 ∈ (◡𝐹 “ 𝐵))))
5	2, 4	syl9r 73	. . . . 5 ⊢ (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑥 ∈ (◡𝐹 “ 𝐵)))))
6	5	imp31 256	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑥 ∈ (◡𝐹 “ 𝐵)))
7	6	ralbidva 2546	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (◡𝐹 “ 𝐵)))
8	1, 7	bitrd 188	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (◡𝐹 “ 𝐵)))
9		dfss3 3236	. 2 ⊢ (𝐴 ⊆ (◡𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (◡𝐹 “ 𝐵))
10	8, 9	bitr4di 198	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220  ^◡ccnv 4773  dom cdm 4774   “ cima 4777  Fun wfun 5371  ‘cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  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 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385

This theorem is used by:  funimass5  5826  funconstss  5827  fimacnv  5837  rinvf1o  6035  iscnp3  15306  cnpnei  15322  cncnp  15333