Theorem funimass3 7000

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 6999 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 6898	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
2		ssel 3916	. . . . . 6 ⊢ (𝐴 ⊆ dom 𝐹 → (𝑥 ∈ 𝐴 → 𝑥 ∈ dom 𝐹))
3		fvimacnv 6999	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑥 ∈ (◡𝐹 “ 𝐵)))
4	3	ex 412	. . . . . 6 ⊢ (Fun 𝐹 → (𝑥 ∈ dom 𝐹 → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑥 ∈ (◡𝐹 “ 𝐵))))
5	2, 4	syl9r 78	. . . . 5 ⊢ (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑥 ∈ (◡𝐹 “ 𝐵)))))
6	5	imp31 417	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) ∈ 𝐵 ↔ 𝑥 ∈ (◡𝐹 “ 𝐵)))
7	6	ralbidva 3159	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (◡𝐹 “ 𝐵)))
8	1, 7	bitrd 279	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (◡𝐹 “ 𝐵)))
9		dfss3 3911	. 2 ⊢ (𝐴 ⊆ (◡𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (◡𝐹 “ 𝐵))
10	8, 9	bitr4di 289	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  ^◡ccnv 5623  dom cdm 5624   “ cima 5627  Fun wfun 6486  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by:  funimass5  7001  funconstss  7002  fvimacnvALT  7003  r0weon  9925  iscnp3  23219  cnpnei  23239  cnclsi  23247  cncls  23249  cncnp  23255  1stccnp  23437  txcnpi  23583  xkoco2cn  23633  xkococnlem  23634  basqtop  23686  kqnrmlem1  23718  kqnrmlem2  23719  reghmph  23768  nrmhmph  23769  elfm3  23925  rnelfm  23928  symgtgp  24081  tgpconncompeqg  24087  eltsms  24108  ucnprima  24256  plyco0  26167  plyeq0  26186  xrlimcnp  26945  rinvf1o  32718  xppreima  32733  rhmpreimacnlem  34044  cvmliftmolem1  35479  cvmlift2lem9  35509  cvmlift3lem6  35522  mclsppslem  35781