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Mirrors > Home > MPE Home > Th. List > Mathboxes > frrlem7 | Structured version Visualization version GIF version |
Description: Lemma for founded recursion. The domain of 𝐹 is a subclass of 𝐴. (Contributed by Paul Chapman, 21-Apr-2012.) |
Ref | Expression |
---|---|
frrlem6.1 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))} |
frrlem6.2 | ⊢ 𝐹 = ∪ 𝐵 |
Ref | Expression |
---|---|
frrlem7 | ⊢ dom 𝐹 ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frrlem6.2 | . . . 4 ⊢ 𝐹 = ∪ 𝐵 | |
2 | 1 | dmeqi 5325 | . . 3 ⊢ dom 𝐹 = dom ∪ 𝐵 |
3 | dmuni 5334 | . . 3 ⊢ dom ∪ 𝐵 = ∪ 𝑔 ∈ 𝐵 dom 𝑔 | |
4 | 2, 3 | eqtri 2644 | . 2 ⊢ dom 𝐹 = ∪ 𝑔 ∈ 𝐵 dom 𝑔 |
5 | iunss 4561 | . . 3 ⊢ (∪ 𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴 ↔ ∀𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴) | |
6 | frrlem6.1 | . . . 4 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))} | |
7 | 6 | frrlem3 31782 | . . 3 ⊢ (𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴) |
8 | 5, 7 | mprgbir 2927 | . 2 ⊢ ∪ 𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴 |
9 | 4, 8 | eqsstri 3635 | 1 ⊢ dom 𝐹 ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 {cab 2608 ∀wral 2912 ⊆ wss 3574 ∪ cuni 4436 ∪ ciun 4520 dom cdm 5114 ↾ cres 5116 Predcpred 5679 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 |
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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-ov 6653 |
This theorem is referenced by: (None) |
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