Theorem funimass3 5405

Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by Raph Levien, 20-Nov-2006.) He remarks: "Likely this could be proved directly, and fvimacnv 5404 would be the special case of A being a singleton, but it works this way round too."

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 5369	. . 3 ⊢ ((Fun F ∧ A ⊆ dom F) → ((F “ A) ⊆ B ↔ ∀x ∈ A (F ‘x) ∈ B))
2		ssel 3268	. . . . . 6 ⊢ (A ⊆ dom F → (x ∈ A → x ∈ dom F))
3		fvimacnv 5404	. . . . . . 7 ⊢ ((Fun F ∧ x ∈ dom F) → ((F ‘x) ∈ B ↔ x ∈ (◡F “ B)))
4	3	ex 423	. . . . . 6 ⊢ (Fun F → (x ∈ dom F → ((F ‘x) ∈ B ↔ x ∈ (◡F “ B))))
5	2, 4	syl9r 67	. . . . 5 ⊢ (Fun F → (A ⊆ dom F → (x ∈ A → ((F ‘x) ∈ B ↔ x ∈ (◡F “ B)))))
6	5	imp31 421	. . . 4 ⊢ (((Fun F ∧ A ⊆ dom F) ∧ x ∈ A) → ((F ‘x) ∈ B ↔ x ∈ (◡F “ B)))
7	6	ralbidva 2631	. . 3 ⊢ ((Fun F ∧ A ⊆ dom F) → (∀x ∈ A (F ‘x) ∈ B ↔ ∀x ∈ A x ∈ (◡F “ B)))
8	1, 7	bitrd 244	. 2 ⊢ ((Fun F ∧ A ⊆ dom F) → ((F “ A) ⊆ B ↔ ∀x ∈ A x ∈ (◡F “ B)))
9		dfss3 3264	. 2 ⊢ (A ⊆ (◡F “ B) ↔ ∀x ∈ A x ∈ (◡F “ B))
10	8, 9	syl6bbr 254	1 ⊢ ((Fun F ∧ A ⊆ dom F) → ((F “ A) ⊆ B ↔ A ⊆ (◡F “ B)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710  ∀wral 2615   ⊆ wss 3258   “ cima 4723  ^◡ccnv 4772  dom cdm 4773  Fun wfun 4776   ‘cfv 4782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-fv 4796

This theorem is referenced by:  funimass5  5406  funconstss  5407  fimacnv  5412