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Mirrors > Home > NFE Home > Th. List > fvco2 | GIF version |
Description: Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 9-Oct-2004.) (Revised by set.mm contributors, 22-Oct-2011.) |
Ref | Expression |
---|---|
fvco2 | ⊢ ((G Fn A ∧ C ∈ A) → ((F ∘ G) ‘C) = (F ‘(G ‘C))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnsnfv 5374 | . . . . . . 7 ⊢ ((G Fn A ∧ C ∈ A) → {(G ‘C)} = (G “ {C})) | |
2 | 1 | imaeq2d 4943 | . . . . . 6 ⊢ ((G Fn A ∧ C ∈ A) → (F “ {(G ‘C)}) = (F “ (G “ {C}))) |
3 | imaco 5087 | . . . . . 6 ⊢ ((F ∘ G) “ {C}) = (F “ (G “ {C})) | |
4 | 2, 3 | syl6reqr 2404 | . . . . 5 ⊢ ((G Fn A ∧ C ∈ A) → ((F ∘ G) “ {C}) = (F “ {(G ‘C)})) |
5 | 4 | eqeq1d 2361 | . . . 4 ⊢ ((G Fn A ∧ C ∈ A) → (((F ∘ G) “ {C}) = {y} ↔ (F “ {(G ‘C)}) = {y})) |
6 | 5 | abbidv 2468 | . . 3 ⊢ ((G Fn A ∧ C ∈ A) → {y ∣ ((F ∘ G) “ {C}) = {y}} = {y ∣ (F “ {(G ‘C)}) = {y}}) |
7 | 6 | unieqd 3903 | . 2 ⊢ ((G Fn A ∧ C ∈ A) → ∪{y ∣ ((F ∘ G) “ {C}) = {y}} = ∪{y ∣ (F “ {(G ‘C)}) = {y}}) |
8 | df-iota 4340 | . . 3 ⊢ (℩zC(F ∘ G)z) = ∪{y ∣ {z ∣ C(F ∘ G)z} = {y}} | |
9 | df-fv 4796 | . . 3 ⊢ ((F ∘ G) ‘C) = (℩zC(F ∘ G)z) | |
10 | imasn 5019 | . . . . . 6 ⊢ ((F ∘ G) “ {C}) = {z ∣ C(F ∘ G)z} | |
11 | 10 | eqeq1i 2360 | . . . . 5 ⊢ (((F ∘ G) “ {C}) = {y} ↔ {z ∣ C(F ∘ G)z} = {y}) |
12 | 11 | abbii 2466 | . . . 4 ⊢ {y ∣ ((F ∘ G) “ {C}) = {y}} = {y ∣ {z ∣ C(F ∘ G)z} = {y}} |
13 | 12 | unieqi 3902 | . . 3 ⊢ ∪{y ∣ ((F ∘ G) “ {C}) = {y}} = ∪{y ∣ {z ∣ C(F ∘ G)z} = {y}} |
14 | 8, 9, 13 | 3eqtr4i 2383 | . 2 ⊢ ((F ∘ G) ‘C) = ∪{y ∣ ((F ∘ G) “ {C}) = {y}} |
15 | df-iota 4340 | . . 3 ⊢ (℩z(G ‘C)Fz) = ∪{y ∣ {z ∣ (G ‘C)Fz} = {y}} | |
16 | df-fv 4796 | . . 3 ⊢ (F ‘(G ‘C)) = (℩z(G ‘C)Fz) | |
17 | imasn 5019 | . . . . . 6 ⊢ (F “ {(G ‘C)}) = {z ∣ (G ‘C)Fz} | |
18 | 17 | eqeq1i 2360 | . . . . 5 ⊢ ((F “ {(G ‘C)}) = {y} ↔ {z ∣ (G ‘C)Fz} = {y}) |
19 | 18 | abbii 2466 | . . . 4 ⊢ {y ∣ (F “ {(G ‘C)}) = {y}} = {y ∣ {z ∣ (G ‘C)Fz} = {y}} |
20 | 19 | unieqi 3902 | . . 3 ⊢ ∪{y ∣ (F “ {(G ‘C)}) = {y}} = ∪{y ∣ {z ∣ (G ‘C)Fz} = {y}} |
21 | 15, 16, 20 | 3eqtr4i 2383 | . 2 ⊢ (F ‘(G ‘C)) = ∪{y ∣ (F “ {(G ‘C)}) = {y}} |
22 | 7, 14, 21 | 3eqtr4g 2410 | 1 ⊢ ((G Fn A ∧ C ∈ A) → ((F ∘ G) ‘C) = (F ‘(G ‘C))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 {cab 2339 {csn 3738 ∪cuni 3892 ℩cio 4338 class class class wbr 4640 ∘ ccom 4722 “ cima 4723 Fn wfn 4777 ‘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: fvco 5384 fvco3 5385 |
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