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Mirrors > Home > MPE Home > Th. List > divsfval | Structured version Visualization version GIF version |
Description: Value of the function in qusval 17425. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by AV, 12-Jul-2024.) |
Ref | Expression |
---|---|
ercpbl.r | ⊢ (𝜑 → ∼ Er 𝑉) |
ercpbl.v | ⊢ (𝜑 → 𝑉 ∈ 𝑊) |
ercpbl.f | ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) |
Ref | Expression |
---|---|
divsfval | ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ercpbl.v | . . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝑊) | |
2 | ercpbl.r | . . . . . 6 ⊢ (𝜑 → ∼ Er 𝑉) | |
3 | 2 | ecss 8695 | . . . . 5 ⊢ (𝜑 → [𝐴] ∼ ⊆ 𝑉) |
4 | 1, 3 | ssexd 5282 | . . . 4 ⊢ (𝜑 → [𝐴] ∼ ∈ V) |
5 | eceq1 8687 | . . . . 5 ⊢ (𝑥 = 𝐴 → [𝑥] ∼ = [𝐴] ∼ ) | |
6 | ercpbl.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
7 | 5, 6 | fvmptg 6947 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ [𝐴] ∼ ∈ V) → (𝐹‘𝐴) = [𝐴] ∼ ) |
8 | 4, 7 | sylan2 594 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝜑) → (𝐹‘𝐴) = [𝐴] ∼ ) |
9 | 8 | expcom 415 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = [𝐴] ∼ )) |
10 | 6 | dmeqi 5861 | . . . . . . . 8 ⊢ dom 𝐹 = dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) |
11 | 2 | ecss 8695 | . . . . . . . . . . 11 ⊢ (𝜑 → [𝑥] ∼ ⊆ 𝑉) |
12 | 1, 11 | ssexd 5282 | . . . . . . . . . 10 ⊢ (𝜑 → [𝑥] ∼ ∈ V) |
13 | 12 | ralrimivw 3148 | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V) |
14 | dmmptg 6195 | . . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V → dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = 𝑉) | |
15 | 13, 14 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → dom (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = 𝑉) |
16 | 10, 15 | eqtrid 2789 | . . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝑉) |
17 | 16 | eleq2d 2824 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝑉)) |
18 | 17 | notbid 318 | . . . . 5 ⊢ (𝜑 → (¬ 𝐴 ∈ dom 𝐹 ↔ ¬ 𝐴 ∈ 𝑉)) |
19 | ndmfv 6878 | . . . . 5 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
20 | 18, 19 | syl6bir 254 | . . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ∅)) |
21 | ecdmn0 8696 | . . . . . 6 ⊢ (𝐴 ∈ dom ∼ ↔ [𝐴] ∼ ≠ ∅) | |
22 | erdm 8659 | . . . . . . . . 9 ⊢ ( ∼ Er 𝑉 → dom ∼ = 𝑉) | |
23 | 2, 22 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → dom ∼ = 𝑉) |
24 | 23 | eleq2d 2824 | . . . . . . 7 ⊢ (𝜑 → (𝐴 ∈ dom ∼ ↔ 𝐴 ∈ 𝑉)) |
25 | 24 | biimpd 228 | . . . . . 6 ⊢ (𝜑 → (𝐴 ∈ dom ∼ → 𝐴 ∈ 𝑉)) |
26 | 21, 25 | biimtrrid 242 | . . . . 5 ⊢ (𝜑 → ([𝐴] ∼ ≠ ∅ → 𝐴 ∈ 𝑉)) |
27 | 26 | necon1bd 2962 | . . . 4 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → [𝐴] ∼ = ∅)) |
28 | 20, 27 | jcad 514 | . . 3 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → ((𝐹‘𝐴) = ∅ ∧ [𝐴] ∼ = ∅))) |
29 | eqtr3 2763 | . . 3 ⊢ (((𝐹‘𝐴) = ∅ ∧ [𝐴] ∼ = ∅) → (𝐹‘𝐴) = [𝐴] ∼ ) | |
30 | 28, 29 | syl6 35 | . 2 ⊢ (𝜑 → (¬ 𝐴 ∈ 𝑉 → (𝐹‘𝐴) = [𝐴] ∼ )) |
31 | 9, 30 | pm2.61d 179 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = [𝐴] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∀wral 3065 Vcvv 3446 ∅c0 4283 ↦ cmpt 5189 dom cdm 5634 ‘cfv 6497 Er wer 8646 [cec 8647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fv 6505 df-er 8649 df-ec 8651 |
This theorem is referenced by: ercpbllem 17431 qusaddvallem 17434 qusgrp2 18866 frgpmhm 19548 frgpup3lem 19560 qusring2 20047 qusrhm 20710 |
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