Mathbox for Jonathan Ben-Naim |
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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 6794 | . . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦))) | |
5 | 4 | necon3abid 3049 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 ≠ 𝐺 ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦))) |
6 | df-ne 3014 | . . . . . . 7 ⊢ ((𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ¬ (𝐹‘𝑦) = (𝐺‘𝑦)) | |
7 | 6 | rexbii 3244 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ∃𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) = (𝐺‘𝑦)) |
8 | rexnal 3235 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 ¬ (𝐹‘𝑦) = (𝐺‘𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦)) | |
9 | 7, 8 | bitri 276 | . . . . 5 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦) ↔ ¬ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐺‘𝑦)) |
10 | 5, 9 | syl6bbr 290 | . . . 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 233 | . 2 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦)) |
13 | nfv 1906 | . . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) ≠ (𝐺‘𝑥) | |
14 | bnj1542.4 | . . . . . 6 ⊢ (𝑤 ∈ 𝐹 → ∀𝑥 𝑤 ∈ 𝐹) | |
15 | 14 | nfcii 2962 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
16 | nfcv 2974 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
17 | 15, 16 | nffv 6673 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
18 | nfcv 2974 | . . . 4 ⊢ Ⅎ𝑥(𝐺‘𝑦) | |
19 | 17, 18 | nfne 3116 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) ≠ (𝐺‘𝑦) |
20 | fveq2 6663 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
21 | fveq2 6663 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐺‘𝑥) = (𝐺‘𝑦)) | |
22 | 20, 21 | neeq12d 3074 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ (𝐹‘𝑦) ≠ (𝐺‘𝑦))) |
23 | 13, 19, 22 | cbvrexw 3440 | . 2 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) ≠ (𝐺‘𝑦)) |
24 | 12, 23 | sylibr 235 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) ≠ (𝐺‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∀wal 1526 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∃wrex 3136 Fn wfn 6343 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 |
This theorem is referenced by: bnj1523 32240 |
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