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Mirrors > Home > MPE Home > Th. List > Mathboxes > elmapintab | Structured version Visualization version GIF version |
Description: Two ways to say a set is an element of mapped intersection of a class. Here 𝐹 maps elements of 𝐶 to elements of ∩ {𝑥 ∣ 𝜑} or 𝑥. (Contributed by RP, 19-Aug-2020.) |
Ref | Expression |
---|---|
elmapintab.1 | ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ (𝐹‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑})) |
elmapintab.2 | ⊢ (𝐴 ∈ 𝐸 ↔ (𝐴 ∈ 𝐶 ∧ (𝐹‘𝐴) ∈ 𝑥)) |
Ref | Expression |
---|---|
elmapintab | ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapintab.1 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ (𝐹‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑})) | |
2 | fvex 6201 | . . . 4 ⊢ (𝐹‘𝐴) ∈ V | |
3 | 2 | elintab 4487 | . . 3 ⊢ ((𝐹‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥)) |
4 | 3 | anbi2i 730 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ (𝐹‘𝐴) ∈ ∩ {𝑥 ∣ 𝜑}) ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥))) |
5 | elmapintab.2 | . . . . . 6 ⊢ (𝐴 ∈ 𝐸 ↔ (𝐴 ∈ 𝐶 ∧ (𝐹‘𝐴) ∈ 𝑥)) | |
6 | 5 | baibr 945 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → ((𝐹‘𝐴) ∈ 𝑥 ↔ 𝐴 ∈ 𝐸)) |
7 | 6 | imbi2d 330 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → ((𝜑 → (𝐹‘𝐴) ∈ 𝑥) ↔ (𝜑 → 𝐴 ∈ 𝐸))) |
8 | 7 | albidv 1849 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥) ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸))) |
9 | 8 | pm5.32i 669 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → (𝐹‘𝐴) ∈ 𝑥)) ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸))) |
10 | 1, 4, 9 | 3bitri 286 | 1 ⊢ (𝐴 ∈ 𝐵 ↔ (𝐴 ∈ 𝐶 ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 ∈ wcel 1990 {cab 2608 ∩ cint 4475 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-pr 4180 df-uni 4437 df-int 4476 df-iota 5851 df-fv 5896 |
This theorem is referenced by: elcnvintab 37908 |
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