![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > rexima | GIF version |
Description: Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Ref | Expression |
---|---|
rexima.x | ⊢ (x = (𝐹‘y) → (φ ↔ ψ)) |
Ref | Expression |
---|---|
rexima | ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (∃x ∈ (𝐹 “ B)φ ↔ ∃y ∈ B ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel2 2934 | . . . 4 ⊢ ((B ⊆ A ∧ y ∈ B) → y ∈ A) | |
2 | funfvex 5135 | . . . . 5 ⊢ ((Fun 𝐹 ∧ y ∈ dom 𝐹) → (𝐹‘y) ∈ V) | |
3 | 2 | funfni 4942 | . . . 4 ⊢ ((𝐹 Fn A ∧ y ∈ A) → (𝐹‘y) ∈ V) |
4 | 1, 3 | sylan2 270 | . . 3 ⊢ ((𝐹 Fn A ∧ (B ⊆ A ∧ y ∈ B)) → (𝐹‘y) ∈ V) |
5 | 4 | anassrs 380 | . 2 ⊢ (((𝐹 Fn A ∧ B ⊆ A) ∧ y ∈ B) → (𝐹‘y) ∈ V) |
6 | fvelimab 5172 | . . 3 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (x ∈ (𝐹 “ B) ↔ ∃y ∈ B (𝐹‘y) = x)) | |
7 | eqcom 2039 | . . . 4 ⊢ ((𝐹‘y) = x ↔ x = (𝐹‘y)) | |
8 | 7 | rexbii 2325 | . . 3 ⊢ (∃y ∈ B (𝐹‘y) = x ↔ ∃y ∈ B x = (𝐹‘y)) |
9 | 6, 8 | syl6bb 185 | . 2 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (x ∈ (𝐹 “ B) ↔ ∃y ∈ B x = (𝐹‘y))) |
10 | rexima.x | . . 3 ⊢ (x = (𝐹‘y) → (φ ↔ ψ)) | |
11 | 10 | adantl 262 | . 2 ⊢ (((𝐹 Fn A ∧ B ⊆ A) ∧ x = (𝐹‘y)) → (φ ↔ ψ)) |
12 | 5, 9, 11 | rexxfr2d 4163 | 1 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (∃x ∈ (𝐹 “ B)φ ↔ ∃y ∈ B ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ∃wrex 2301 Vcvv 2551 ⊆ wss 2911 “ cima 4291 Fn wfn 4840 ‘cfv 4845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-fv 4853 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |