| Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1542 | Structured version Visualization version GIF version | ||
| Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bnj1542.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
| bnj1542.2 | ⊢ (𝜑 → 𝐺 Fn 𝐴) |
| bnj1542.3 | ⊢ (𝜑 → 𝐹 ≠ 𝐺) |
| bnj1542.4 | ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹) |
| Ref | Expression |
|---|---|
| bnj1542 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bnj1542.3 | . . 3 ⊢ (𝜑 → 𝐹 ≠ 𝐺) | |
| 2 | bnj1542.1 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
| 3 | bnj1542.2 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐴) | |
| 4 | eqfnfv 7051 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦))) | |
| 5 | 4 | necon3abid 2977 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ≠ 𝐺 ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦))) |
| 6 | df-ne 2941 | . . . . . . 7 ⊢ ((𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ¬ (𝐹‘𝑦) = (𝐺‘𝑦)) | |
| 7 | 6 | rexbii 3094 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ∃𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) = (𝐺‘𝑦)) |
| 8 | rexnal 3100 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) = (𝐺‘𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦)) | |
| 9 | 7, 8 | bitri 275 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦)) |
| 10 | 5, 9 | bitr4di 289 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ≠ 𝐺 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦))) |
| 11 | 2, 3, 10 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐹 ≠ 𝐺 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦))) |
| 12 | 1, 11 | mpbid 232 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦)) |
| 13 | nfv 1914 | . . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) ≠ (𝐺‘𝑥) | |
| 14 | bnj1542.4 | . . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹) | |
| 15 | 14 | nfcii 2894 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
| 16 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
| 17 | 15, 16 | nffv 6916 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
| 18 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥(𝐺‘𝑦) | |
| 19 | 17, 18 | nfne 3043 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) ≠ (𝐺‘𝑦) |
| 20 | fveq2 6906 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
| 21 | fveq2 6906 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐺‘𝑥) = (𝐺‘𝑦)) | |
| 22 | 20, 21 | neeq12d 3002 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ (𝐹‘𝑦) ≠ (𝐺‘𝑦))) |
| 23 | 13, 19, 22 | cbvrexw 3307 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦)) |
| 24 | 12, 23 | sylibr 234 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Fn wfn 6556 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: bnj1523 35085 |
| Copyright terms: Public domain | W3C validator |